diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzaxnu" "b/data_all_eng_slimpj/shuffled/split2/finalzzaxnu" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzaxnu" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n\n\tNematic liquid crystal (NLC) is a liquid crystal phase with has rod-shaped molecules which tend to align along a particular direction denoted by a unit vector $\\mathbf{n}$, called the optical director axis. In addition to $\\mathbf{n}:\\mathbb{R}^d \\to \\mathbb{R}^3$ , the hydrodynamic of an isothermal and incompressible NLC is also described by its pressure $p:\\mathbb{R}^d\\to \\mathbb{R}$ and velocity $\\mathbf{v}:\\mathbb{R}^d\\to \\mathbb{R}^d$.\nWe refer to \\cite{Chandrasekhar}\nand \\cite{Gennes} for a comprehensive treatment of the physics of\nliquid crystals.\n\t\nUsing the Ericksen and Leslie continuum theory for liquid crystals, see \\cite{Ericksen} and\nLeslie \\cite{Leslie}, F. Lin and C. Liu\n\\cite{Lin-Liu} derived the most basic and simplest form of the dynamical system modeling the motion of a nematic liquid crystal (NLC)\nflowing in $\\mathbb{R}^d (d=2,3)$. This system is given by \n\t\t\\begin{align}\n\t&\td\\mathbf{v}+\\biggl[(\\mathbf{v}\\cdot\\nabla)\\mathbf{v}-\\Delta \\mathbf{v}+\\nabla p\\biggr]dt= -\\nabla\\cdot(\\nabla \\mathbf{n} \\odot \\nabla \\mathbf{n})dt+ \\mathbf{f} dt,\\label{eqn-SELE-v} \\\\\n\t&\t\\mathrm{div }\\; \\mathbf{v}= 0,\\label{eqn-SELE-div} \\\\\n\t&\td\\mathbf{n}+(\\mathbf{v}\\cdot\\nabla)\\mathbf{n} dt = \\biggl[\\Delta\n\t\\mathbf{n}+ \\lvert \\nabla \\d \\rvert^2 \\d \\biggr]dt+\\mathbf{g} dt , \\label{eqn-SELE-d}\\\\\n\t& \\lvert \\mathbf{n} \\rvert^2= 1,\\label{eqn-SELE-sphere}\n\t\\end{align}\n\twhere $\\mathbf{f}$ and $\\mathbf{g}$ are forcings acting on the system.\n\t The entries of the matrix $\\nabla \\mathbf{n} \\odot \n\\nabla \\mathbf{n}$ are defined by \n\\begin{equation*}\n[\\nabla \\mathbf{n} \\odot \\nabla \\mathbf{n}]_{i,j}=\\sum_{k=1}^3 \\frac{\\partial\n\t\\mathbf{n}^{(k)}}{\\partial x_i}\\frac{\\partial \\mathbf{n}^{(k)}}{\\partial x_j},\\;\\;\n\\mbox{ } i,j=1,\\dots, d.\n\\end{equation*}\nBefore proceeding further, we should mention that \\eqref{eqn-SELE-v}-\\eqref{eqn-SELE-sphere} is obtained by neglecting several terms such as the viscous Leslie stress tensor in the equation for $\\mathbf{v}$, the stretching and rotational effects for $\\mathbf{d}$. Thus, it is not known whether the models \\eqref{eqn-SELE-v}-\\eqref{eqn-SELE-sphere} and \\eqref{eqn-SLQE-v}-\\eqref{eqn-SLQE-d} are thermodynamically stable or consistent. However, these models still retain many mathematical and essential features of the dynamics for NLCs. In the recent papers \\cite{MH+JP-2017} \\cite{MH+JP-2018}, \\cite{MH et al-2014}, \\cite{Lin+Wang-2014} and \\cite{Sun+Liu} several thermodynamically consistent and stable models of NLC have been developed and analysed.\n\t\n\tIn this paper, we fix a bounded domain $\\mathcal{O}\\subset \\mathbb{R}^d$, $d=12,3$ with smooth boundary and we consider the following stochastic system\n\t\\begin{align}\n\t&\td\\mathbf{v}+\\Bigl[(\\mathbf{v}\\cdot\\nabla)\\mathbf{v}-\\Delta \\mathbf{v}+\\nabla p\\Bigr]dt= -\\nabla\\cdot(\\nabla \\mathbf{n} \\odot \\nabla \\mathbf{n})dt+ S(\\mathbf{v}) dW_1,\\label{eqn-SLQE-v} \\\\\n\t&\t\\mathrm{div }\\; \\mathbf{v}= 0,\\label{eqn-SLQE-div} \\\\\n\t&\td\\mathbf{n}+(\\mathbf{v}\\cdot\\nabla)\\mathbf{n} dt = \\bigl[\\Delta\n\t\\mathbf{n}+f(\\d)\\bigr]dt+(\\mathbf{n}\\times \\mathbf{h})\\circ\n\tdW_2, \\label{eqn-SLQE-d}\\\\\n\t&\t\\mathbf{v}=0 \\text{ and } \\frac{\\partial \\mathbf{n}}{\\partial \\boldsymbol{\\nu}}=0 \\text{ on }\n\t\\partial \\mathcal{O},\\label{BC}\\\\\n& \\mathbf{v}(0)=\\mathbf{v}_0 \\text{ and } \\mathbf{n}(0)=\\mathbf{n}_0, \\label{Init-Cond}\n\t\\end{align}\n\twhere $\\mathbf{v}_0: \\mathcal{O} \\to \\mathbb{R}^d$, $\\mathbf{n}_0:\\mathcal{O}\\to \\mathbb{R}^3$ are given mappings, $\\boldsymbol{\\nu}$ is the unit outward normal to $\\partial \\mathcal{O}$, $f$ is a polynomial function satisfying some conditions to be fixed later. Here, $W_1$ and $W_2$ are respectively independent cylindrical Wiener process and standard Brownian motion, $(\\mathbf{n}\\times \\mathbf{h})\\circ dW_2$ is understood in\n\tthe Stratonovich sense.\n\n\n\tThe Fr\\'eedericksz transition, which is produced by applying a sufficiently strong external perturbation (e.g. magnetic or electric fields) to an undistorted NLC, and its behaviour under random perturbation have been extensively studied in several physics papers, see \\cite{Horsthemke+Lefever-1984, San Miguel-1985,FS+MSanM}, all of which neglected the fluid velocity. However, it is pointed out in \\cite[Chapter 5]{Gennes} that the fluid flow disturbs the alignment and conversely a change in the alignment will induce a flow in the nematic liquid crystal. It is this gap in knowledge that is the motivation for our mathematical study which was initiated in the old unpublished preprints \\cite{BHP13} and \\cite{BHP-arxiv}, see also the recent papers \\cite{BHP18} and \\cite{BHP19}.\n\n\n\t\n\tIn this paper, we mainly prove the existence and uniqueness of a maximal local strong solution which is\nunderstood in the sense of stochastic calculus and PDEs. This result is a corollary of several abstract results which are proved in Section \\ref{ABST-STRONG} and are of independent interest. \n In the case $d=2$, we show the non-explosion of the maximal solution by an adaptation and combination of Khashminskii test for non-explosions and an idea of Schmalfu{\\ss} elaborated in \\cite{Bjorn}, see Section \\ref{SLC-Sect4} for more details. Our novelty is the extension of the Schmalfu{\\ss} idea, which has been used so far to prove the uniqueness of solutions of Stochastic Navier-Stokes equations and related problems, to the proof of the global existence of a strong solution to the problem \\eqref{eqn-SLQE-v}-\\eqref{Init-Cond}. In particular, we give another proof of the global existence of 2D stochastic Navier-Stokes equations with multiplicative noise and for initial data with finite enstrophy. Thus, our paper can also be seen as a generalization of the results for the existence and uniqueness of maximal local and global solutions of strong solutions of stochastic Navier-Stokes proved in \\cite{Nathan1}, \\cite{Nathan2} and \\cite{Mikulevicius}.\n\nWe should notice that some of the arguments elaborated in Section \\ref{ABST-STRONG} have been already used in \\cite{BHR-2014} and \\cite{BHP19} which respectively studied the strong solution of some stochastic hydrodynamic equations (NSEs, MHD and 3D Leray $\\alpha$-models) driven by L\\'evy noise, and the existence and uniqueness of a maximal local smooth solution to the stochastic Ericksen-Leslie system \\eqref{eqn-SELE-v}-\\eqref{eqn-SELE-sphere} on the $d$-dimensional torus. We are also strongly convinced that with these general results it is possible, although it has not been done in detail, to prove the existence of strong solution of several stochastic hydrodynamical models such as the NSEs, MHD equations, $\\alpha$-models for Navier-Stokes and related problems.\n\n\t\n\tWhile the deterministic version of \\eqref{eqn-SLQE-v}-\\eqref{eqn-SLQE-d} has been the subject of intensive mathematical studies, see \\cite{Rojas-Medar, Hong,Lin-Liu,Lin-Wang,Lin-Wang-2016, Shkoller,Dai+Schonbeck_2014} and \\cite{Cavaterra,Hong-2014,Hong-2012,Huang-2014, Wang}, there are fewer results related to the stochastic system \\eqref{eqn-SLQE-v}-\\eqref{eqn-SLQE-d}. The unpublished paper \\cite{BHP13} proved the existence and uniqueness of mxaimal local strong solution to the system \\eqref{eqn-SLQE-v}-\\eqref{eqn-SLQE-d} with a bounded nonlinear term $f(\\d)=\\mathds{1}_{\\lvert \\d \\rvert\\le 1} (1-\\lvert \\d\\rvert^2)\\d $. The paper \\cite{BHP18} deals only with weak (both in PDEs and stochastic calculus sense) solutions and the maximum principle. Some of the results in \\cite{BHP18} and the current paper have already been used in several papers such as \\cite{ZB+UM+AP}, \\cite{ZB+UM+AP-2019}, \\cite{RZ+GZ}, \\cite{BGuo+GZhou-2019}, \\cite{GZhou-2019} and \\cite{Wang+Wu+Zhou-2019}.\n\n\t\n\t Very recently we have become aware of a recent paper by Feireisl and Petcu \\cite{FeirPetcu_2019}, in which they proved the existence of a \\textit{dissipative martingale}, as well as the existence of a local strong solution and weak-strong uniqueness of the solution of the stochastic Navier-Stokes Allen-Cahn Equations. Note that in \\cite{FeirPetcu_2019} the second unknown $\\mathbf{n}$ is a scalar field, the nonlinear term $f(\\cdot)$ is globally Lipschitz and the derivative of a double-well potential $F(\\cdot)$, and the coefficient of the noise entering the equations for $\\mathbf{n}$ is bounded.\n\tThe paper \\cite{BHP19} is the first paper to deal with the the stochastic counterpart of the Ericksen-Leslie equations \\eqref{eqn-SELE-v}-\\eqref{eqn-SELE-sphere}. The results in present manuscript is not covered in \\cite{BHP19} because in contrast to our framework which considers initial condition $(\\v_0, \\d_0)\\in \\mathrm{H}^1 \\times \\mathbf{H}^{2}$, the initial data in \\cite{BHP19} satisfies $(\\v_0, \\d_0)\\in \\mathrm{H}^{\\alpha}\\times \\mathrm{H}^{\\alpha+1}$ for $\\alpha>\\frac d2$, where $d=2,3$ is the space dimension. There is also the papers \\cite{Medjo1} which seeks for a special solution $(\\mathbf{v},\\mathbf{n})$ with the unknown $\\mathbf{n}$ is replaced by an angle $\\theta$ such that $\\mathbf{n}=(\\cos\\theta, \\sin\\theta)$. This model reduction considerably simplify the mathematical analysis of \\eqref{eqn-SELE-v}-\\eqref{eqn-SELE-d}.\n\t\n\t\n\tTo close this introduction, we emphasize that the analysis in the present paper might also be of great interest in the numerical study of stochastic Ericksen-Leslie system. In fact, on the one hand our assumptions on the polynomial $f(\\d)$ enable us to consider the typical Ginzburg-Landau function $f_\\varepsilon(\\d)=\\frac{1}{\\varepsilon^2}(1 -\\lvert \\d \\rvert^2) \\d$, see Assumption \\ref{eqn-f} and Remark \\ref{Rem:Ginz-Land-General}. In the numerical context, handling the constraint $\\lvert \\d\\rvert=1$ in the Ericksen-Leslie is a rather challenging task and to overcome this difficulty, one usually use the Ginzburg-Landau approximation, see \\cite{Walkington}. On the other hand, to get convergence and a rate of convergence of a space discretization for parabolic SPDEs, one often has to consider a regular solution in favour of weak solutions. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\\section{Preliminary results and notations}\\label{sec-spaces-2}\n\n\t\\subsection{Functional spaces and linear operators}\n\tFollowing \\cite{BHP18} we introduce in this section various notations and results that are frequently used in this paper.\n\t\n\t For two topological spaces $X$ and $Y$ the symbol $X\\hookrightarrow Y$ means that the embedding $X$ is continuously embedded in $Y$. \n\t\n\t\\noindent Let $d\\in \\{2,3\\}$ and assume that $\\mathcal{O} \\subset \\mathbb{R}^d$ is a bounded domain with boundary $\\partial \\mathcal{O}$ of class $\\mathcal{C}^\\infty$.\n\tFor $p\\in [1,\\infty)$ and $k\\in \\mathbb{N}$ the symbols $\\mathrm{L}^p(\\mathcal{O})$ or by $\\mathbb{W}^{k,p}(\\mathcal{O})$ (resp. by $\\mathbf{L}^p(\\mathcal{O})$ or by $\\mathbf{W}^{k,p}(\\mathcal{O})$) respectively denote the Lebesgue and Sobolev spaces of functions $\\mathbf{v}:\\mathbb{R}^d\\to \\mathbb{R}^d$ (resp. $\\mathbf{n}:\\mathbb{R}^d \\to \\mathbb{R}^3$) \n\tFor $p=2$ the function spaces $\\mathbb{W}^{k,2}(\\mathcal{O})$ and $\\mathbf{W}^{k,p}$ are respectively denoted by $\\mathrm{H}^k$ ($\\mathbf{H}^k$) and their norms are \n\tdenoted by $\\lVert \\cdot\n\t\\rVert_k$. The scalar products on $\\mathrm{L}^2$ and $\\mathbf{L}^2$ are denoted by the same symbol $\\langle\n\tu,v\\rangle$ for $u,v\\in \\mathrm{L}^2$ (resp. $u,v\\in \\mathbf{L}^2$ ) and their associated norms is denoted by $\\lVert\n\tu\\rVert$, $u\\in \\mathrm{L}^2$ (resp. $u\\in \\mathrm{L}^2$).\n\tBy $\\mathrm{H}^1_0$ and $\\mathrm{W}^{1,r}_0$, $r>2$, we mean the spaces of functions in $\\mathrm{H}^1$ and $\\mathrm{W}^{1,r}$\n\tthat vanish on the boundary on $\\mathcal{O}$. \n\t It is well known that if $a=\\frac{d}4$, then there exists a constants $c>0$ such that \n\t\\begin{equation} \\label{GAG-l4}\n\t\\lvert \\mathbf{u}\\rvert_{\\mathrm{L}^4}\\le c \\begin{cases}\n \\lvert \\mathbf{u}\\rvert_{\\mathrm{L}^2}^{1-a} \\lvert \\nabla \\mathbf{u}\n\\rvert^a_{\\mathrm{L}^2} \t\\text{ if } \\mathbf{u}\\in \\mathbb{H}^{1}_{0}\\\\\n\\lvert \\mathbf{u}\\rvert_{\\mathrm{L}^2}^{1-a} \\lvert \\mathbf{u}\n\\rvert^a_{\\mathbb{H}^1} \t\\text{ if } \\mathbf{u}\\in \\mathbb{H}^{1},\n\t\\end{cases}\n\t\\end{equation}\n\t\t\\begin{equation} \\label{GAG-LInf}\n\t\\lvert \\mathbf{u}\\rvert_{\\mathrm{L}^\\infty}\\le c \\begin{cases}\n\t\\lvert \\mathbf{u}\\rvert_{\\mathrm{L}^4}^{1-a} \\lvert \\nabla \\mathbf{u}\n\t\\rvert^a_{\\mathrm{L}^4} \t\\text{ if } \\mathbf{u}\\in \\mathbb{W}^{1,4}_{0}\\\\\n\t\\lvert \\mathbf{u}\\rvert_{\\mathrm{L}^4}^{1-a} \\lvert \\mathbf{u}\n\t\\rvert^a_{\\mathbb{W}^{1,4}} \t\\text{ if } \\mathbf{u}\\in \\mathbb{W}^{1,4},\n\t\\end{cases}\n\t\\end{equation}\n\tand since $\\mathrm{H}^{2} \\hookrightarrow \\mathrm{W}^{1,4}$, we have \n\t\\begin{equation}\n\t\\lvert \\mathbf{u} \\rvert_{\\mathrm{L}^\\infty}\\le \\lVert \\mathbf{u} \\rVert_1 ^{1-a}\\lVert\n\t\\mathbf{u} \\rVert_{2}^a, \\; \\mathbf{u}\\in \\mathrm{H}^2 .\\label{GAG-LInf-2}\n\t\\end{equation}\n\t\n\tWe now introduce the following spaces\n\t\\begin{align*}\n\t\\mathcal{V}& =\\left\\{ \\mathbf{u}\\in \\mathcal{C}_{c}^{\\infty }(\\mathcal{O},\\mathbb{R}^d)\\,\\,\\text{such that}%\n\t\\,\\,\\Div \\mathbf{u}=0\\right\\} \\\\\n\t\\mathrm{V}& =\\,\\,\\text{closure of $\\mathcal{V}$ in }\\,\\,\\mathrm{H}_0^{1}(\\mathcal{O}) \\\\\n\t\\mathrm{H}& =\\,\\,\\text{closure of $\\mathcal{V}$ in\n\t}\\,\\,\\mathrm{L}^{2}(\\mathcal{O}).\n\t\\end{align*}\n\tAs usual, we endow $\\mathrm{H}$ with the scalar product and norm of $\\mathrm{L}^2$, and we equip the space $\\mathrm{V}$ with he scalar product\n\t$\\langle \\nabla \\mathbf{u}, \\nabla \\mathbf{v}\\rangle$ which is equivalent to the\n\t$\\mathrm{H}^1(\\mathcal{O})$-scalar product on $\\mathrm{V}$.\n\t\n\tLet $\\Pi: \\mathrm{L}^2 \\rightarrow \\mathrm{H}$ be the Helmholtz-Leray projection\n\tfrom $\\mathrm{L}^2$ onto $\\mathrm{H}$. We denote by $\\mathrm{A}=-\\Pi\\Delta$ the\n\tStokes operator with domain $D(\\mathrm{A})=\\mathrm{V}\\cap \\mathrm{H}^2$, see for instance \\cite[Chapter I, Section 2.6]{Temam} . It is well-known that the spaces $\\mathrm{V}_\\beta:=D(\\mathrm{A}^\\beta)$, $\\beta \\in \\mathbb{R}$, are Hilbert spaces when endowed with the graph inner product and $\\mathrm{V}_\\frac12 = \\mathrm{V}$. It is also well-known that the map $\\mathrm{A}^\\delta: \\mathrm{V}_\\beta \\to \\mathrm{V}_{\\beta-\\delta}$, $\\beta,\\delta \\in \\mathbb{R}$, is a linear isomorphism. For all these facts we refer, for instance, to \\cite{PC+CF}.\n\n\t\n The Neumann Laplacian acting on $\\mathbb{R}^d$-valued\n\tfunction will be denoted by $\\mathrm{A}_1$, that is,\n\t\\begin{equation*}\n\t\\begin{split}\n\tD(\\mathrm{A}_1)&:=\\biggl\\{\\mathbf{u}\\in \\mathbf{H}^2: \\frac{\\partial \\mathbf{u}}{\\partial \\boldsymbol{\\nu}}=0 \\text{ on } \\partial \\mathcal{O}\\biggr\\},\\\\\n\t\\mathrm{A}_1\\mathbf{u}&:=-\\sum_{i=1}^d \\frac{\\partial^2 \\mathbf{u}}{\\partial\n\t\tx_i^2},\\;\\;\\; \\mathbf{u} \\in D(\\mathrm{A}_1).\n\t\\end{split}\n\t\\end{equation*}\n\tNotice that the Neumann Laplacian $\\mathrm{A}_1$ can be viewed as a linear map $\\mathrm{A}_1:\\mathbf{H}^1 \\to (\\mathbf{H}^1)^\\ast$ satisfying \t\n\t\\begin{equation}\\label{Eq:WeakNeumannLaplace}\n\t{}_{(\\mathbf{H}^1)^\\ast}\\langle \\mathrm{A}_1 \\mathbf{u}, \\mathbf{n}\\rangle_{\\mathbf{H}^1}=\\langle \\nabla \\mathbf{u}, \\nabla \\mathbf{n} \\rangle,\\text{ for all } \\mathbf{u},\\mathbf{n} \\in \\mathbf{H}^1.\n\t\\end{equation}\n\tThanks to \\cite[Theorem 5.31]{Haroske} one can define and characterize in standard way the spaces $\\mathbf{X}_\\alpha=D(\\hat{\\mathrm{A}}_1^{\\alpha})$, $\\alpha \\in [0,\\infty)$, where $\\hat{\\mathrm{A}}_1=I+\\mathrm{A}_1$. Also, it can be shown that $\\mathbf{X}_\\alpha \\hookrightarrow \\mathbf{H}^{2\\alpha}$, for all $\\alpha \\geq 0$ and\n\t$\\mathbf{X}:=\\mathbf{X}_{\\frac12}=\\mathbf{H}^1$, see, for instance, \\cite[Sections 4.3.3 \\& 4.9.2]{Triebel}. \n\n\t\n\t\n\tNow, let $\\mathbf{h}\\in\\mathbf{L}^\\infty$ be fixed and define a linear bounded operator $G$ from $\\mathbf{L}^2$ into itself by\n\\begin{equation}\\label{Eq:LinG}\n\t G: \\mathbf{L}^2 \\ni \\mathbf{n} \\mapsto \\mathbf{n}\\times \\mathbf{h}\\in \\mathbf{L}^2.\n\\end{equation}It is straightforward to check that there exists a constant $C>0$ such that\n\t$$ \\lVert G(\\mathbf{n})\\rVert \\le C \\lVert \\mathbf{h}\\rVert_{\\mathbf{L}^\\infty} \\lvert \\mathbf{n} \\rvert_{\\mathbf{L}^2}, \\text{ for all } \\mathbf{n} \\in \\mathbf{L}^2.$$\n\t\n\t\\subsection{The nonlinear terms}\n\t\n\tThroughout this paper $\\mathbf{B}^\\ast$ denotes the dual space of\n\ta Banach space $\\mathbf{B}$. We also denote by $\\langle \\Psi,\n\t\\mathbf{b}\\rangle_{\\mathbf{B}^\\ast,\\mathbf{B}}$ the value of $\\Psi\\in \\mathbf{B}^\\ast$ on\n\t$\\mathbf{b}\\in \\mathbf{B}$. Throughout, $\\partial_{x_i}=\\frac{\\partial}{\\partial x_i} $ and {$\\phi^{(i)}$}\n\tis the $i$-th entry of any vector-valued $\\phi$.\n\t\n\tLet $p,q,r \\in [1,\\infty]$ such that $\\frac 1p +\\frac 1q +\\frac 1r\\le 1.$ Then, we define a trilinear form $b(\\cdot, \\cdot,\\cdot)$ by\n\t\\begin{equation*}\n\tb(\\mathbf{u},\\mathbf{v},\\mathbf{w})=\\sum_{i,j=1}^d \\int_\\mathcal{O}\\mathbf{u}^{(i)}\\frac{\\partial\n\t\t\\mathbf{v}^{(j)}}{\\partial x_i}\\mathbf{w}^{(j)} dx,\\,\\, \\mathbf{u}\\in \\mathrm{L}^p ,\\mathbf{v}\\in\n\t\\mathbb{W}^{1,q}, \\text{ and } \\mathbf{w}\\in \\mathrm{L}^r,\n\t\\end{equation*}\n\tNote if $\\mathbf{v}\\in \\mathbf{W}^{1,q}$ and $\\mathbf{w} \\in \\mathbf{L}^r$, then we have to take the sum over $j$ from $j=1$ to $j=3$.\n\t\n\tIt is well known, see \\cite[Section II.1.2]{Temam}, that there is a bilinear map $B: \\mathrm{V} \\times \\mathrm{V} \\to \\mathrm{V}^\\ast$ such that\n\t\\begin{equation}\\label{DEF-B1}\n\t\\langle\n\tB(\\mathbf{u},\\mathbf{v}),\\mathbf{w}\\rangle_{\\mathrm{V}^\\ast,\\mathrm{V}}=b(\\mathbf{u},\\mathbf{v},\\mathbf{w})\\text{ for } \\mathbf{w}\\in \\mathrm{V},\\text{ and } \\mathbf{u}, \\mathbf{v}\\in \\mathrm{V}.\n\t\\end{equation}\n\tIn a similar way, there is also a bilinear map$\\tilde{B}:\\mathrm{V} \\times \\mathbf{H}^1\\to (\\mathbf{H}^1)^\\ast$\n\tsuch that\n\t\\begin{equation}\\label{DEF-B2}\n\t\\langle\n\t\\tilde{B}(\\mathbf{u},\\mathbf{v}),\\mathbf{w}\\rangle_{(\\mathbf{H}^1)^\\ast,\\mathbf{H}^1}=b(\\mathbf{u},\\mathbf{v},\\mathbf{w})\\,\\, \\text{ for all }\n\t\\mathbf{u} \\in \\mathrm{V}, \\,\\, \\mathbf{v}, \\,\\,\\mathbf{w}\\in \\mathbf{H}^1.\n\t\\end{equation}\n\tThe following lemma was proved in \\cite[Section II.1.2]{Temam}. \n\t\\begin{lem}\\label{LEM-B}\n\t\tThe bilinear map $B(\\cdot, \\cdot)$ maps continuously $\\mathrm{V}\\times\n\t\t\\mathrm{H}^1$ into $\\mathrm{V}^\\ast$ and\n\t\t\\begin{align}\n\t\t&\\langle B(\\mathbf{u},\\mathbf{v}),\\mathbf{w}\\rangle_{\\mathrm{V}^\\ast,\\mathrm{V}}=b(\\mathbf{u},\\mathbf{v},\\mathbf{w}), \\text{ for all }\n\t\t\\mathbf{u}\\in \\mathrm{V}, \\mathbf{v}\\in \\mathrm{H}^1,\\mathbf{w} \\in \\mathrm{V},\\label{B1}\\\\\n\t\t&\\langle B(\\mathbf{u},\\mathbf{v}),\\mathbf{w}\\rangle_{\\mathrm{V}^\\ast,\\mathrm{V}}=-b(\\mathbf{u},\\mathbf{w},\\mathbf{v})\n\t\t\\text{ for all } \\mathbf{u}\\in \\mathrm{V}, \\mathbf{v}\\in \\mathrm{H}^1,\\mathbf{w} \\in \\mathrm{V},\\label{B2}\\\\\n\t\t&\\langle B(\\mathbf{u},\\mathbf{v}),\\mathbf{v}\\rangle_{\\mathrm{V}^\\ast,\\mathrm{V}}=0 \\,\\, \\text{ for all } \\mathbf{u}\\in \\mathrm{V},\n\t\t\\mathbf{v}\\in \\mathrm{V},\\label{B3}\\\\\n\t\t&\\lvert B(\\mathbf{u},\\mathbf{v})\\rvert_{\\mathrm{V}^\\ast}\\le C_0 \\lvert \\mathbf{u}\\rvert_{\\mathrm{L}^2}^{1-\\frac d4}\n\t\t\\lvert\\nabla \\mathbf{u} \\rvert_{\\mathrm{L}^2}^{\\frac d4} \\lvert \\mathbf{v}\\rvert_{\\mathrm{L}^2}^{1-\\frac d4}\\rvert_{\\mathrm{L}^2}\n\t\t\\nabla \\mathbf{v}\\rvert_{\\mathrm{L}^2}^\\frac d4, \\text{ for all } \\mathbf{u}\\in \\mathrm{V}, \\mathbf{v}\\in\n\t\t\\mathrm{H}^1.\\label{B4}\n\t\t\\end{align}\n\t\\end{lem}\n\t\\begin{lem}\\label{LEM-G1}\n\tThere exists a constant $C_1>0$\n\t\tsuch that\n\t\t\\begin{align}\n\t&\t\\lVert \\tilde{B} (\\mathbf{v}, \\mathbf{n})\\lVert \\le C_1 \\lVert \\mathbf{v}\\rVert_{\\mathrm{L}^2}^{1-\\frac\n\t\t\td4}\\rVert \\nabla \\mathbf{v}\\rVert_{\\mathrm{L}^2}^\\frac d4 \\lVert \\mathbf{n}\\rVert_{\\mathbf{H}^1}^{1-\\frac d4}\n\t\t\\lVert \\mathbf{n} \\rVert_{\\mathbf{H}^2}^{\\frac d4}, \\text{ for all } \\mathbf{v} \\in \\mathrm{V},\n\t\t\\mathbf{n}\\in \\mathrm{H}^2,\\label{EST-G1}\\\\\n\t&\t\\langle \\tilde{B}(\\mathbf{v},\\mathbf{n}),\\mathbf{n}\\rangle=0,\\text{\n\t\t\tfor\n\t\t\tany } \\mathbf{v}\\in \\mathrm{V}, \\mathbf{n} \\in \\mathrm{H}^2.\\label{tild-b-0}\n\t\t\\end{align}\n\t\\end{lem}\n\t\\begin{proof}\n\tThe estimate \\eqref{EST-G1} follows from \\cite[Lemma 6.2]{BHP18} and the Gagliardo-Nirenberg estimate \\eqref{GAG-l4}. The proof of \\eqref{tild-b-0} and \\eqref{B3} are the same, see \\cite[Section II.1.2]{Temam}. \n\t\\end{proof}\n\t\n\t\n\tLet $r,\\,p,\\, q\\in(1,\\infty)$ such that $ \\frac 1p\n\t+\\frac 1q+\\frac 1r\\le 1$. For \\text{ $\\d_1\\in \\mathbf{W}^{1,p}$, $\\d_2\\in \\mathbf{W}^{1,q}$ and $\\mathbf{u}\\in\n\t\t\\mathbb{W}^{1,r}$ } we set \n\t\\begin{equation}\\label{INT-md}\n\t\\mathfrak{m}(\\d_1, \\d_2,\\mathbf{u})= -\\sum_{i,j=1}^d\\sum_{k=1}^3 \\int_\\mathcal{O}\n\t\\partial_{x_i}\\mathbf{n}_1^{(k)} \\partial_{x_j}\\mathbf{n}_2^{(k)} \\partial_{x_j}\\mathbf{u}^{(i)}\\,dx. \n\t\\end{equation}\n\tSince $d\\le 4$,\n\tthe integral in \\eqref{INT-md} is also well defined for $\\mathbf{n}_1,\n\t\\d_2 \\in \\mathbf{H}^2$ and $\\mathbf{u}\\in \\mathrm{V}$.\n\t\n\t\n\tWe recall the following proposition which can be found in \\cite[Proposition 2.2 \\& Remark 2.3]{BHP18}.\n\t\\begin{prop}\\label{LEM-M}\n\t\tLet $d\\in [1,4]$. There exists a bilinear map $M:\\mathbf{H}^2\\times \\mathbf{H}^2 \\to \\mathrm{V}^\\ast $ such that\n\t\t\\begin{align}\\label{def-Md}\n\t\t&\\langle M(\\mathbf{n}_1,\\d_2), \\mathbf{u}\\rangle_{\\mathrm{V}^\\ast,\\mathrm{V}}= \\mathfrak{m}(\\d_1,\\d_2,\\mathbf{u}), \\;\\text{$\\d_1, \\, \\d_2 \\in \\mathbf{H}^2$, } \\mathbf{u} \\in \\mathrm{V},\\\\\t\n\t\t&\t\\langle M(\\mathbf{f}, \\mathbf{g}), \\mathbf{v} \\rangle_{\\mathrm{V}^\\ast, \\mathrm{V}}= \\langle \\Pi[\\Div (\\nabla \\mathbf{f}\n\t\t\\odot \\nabla \\mathbf{g})], \\mathbf{v} \\rangle \\text{ for all } \\mathbf{f}, \\mathbf{g}\\in \\mathbf{X}_{1} \\text{ and } \\mathbf{v} \\in \\mathrm{H},\\label{Eq:Identity-M-L2}\\\\\n\t\t&\t\\langle \\tilde{B}(\\mathbf{v},\\mathbf{n}), \\mathrm{A}_1 \\mathbf{n}\\rangle+\\langle\n\t\tM(\\mathbf{n},\\d), \\mathbf{v}\\rangle_{\\mathrm{V}^\\ast,\\mathrm{V}}=0, \\text{ for all } \\mathbf{v}\\in \\mathrm{V}, \\mathbf{n} \\in \\mathbf{X}_1,\\label{G1-eq-Md}.\n\t\t\\end{align}\n\t\\end{prop}\n\tIn some places in this manuscript we use the following shorthand notations:\n\t$$ B(\\mathbf{u}):= B(\\mathbf{u},\\mathbf{u}) \\text{ and } M(\\mathbf{n}):= M(\\mathbf{n},\\mathbf{n}),$$ for all $\\mathbf{u}$ and $\\mathbf{n}$ such that the above quantities are meaningful.\n\n\tWe now fix the standing assumptions on the function $f(\\cdot)$.\n\t\\begin{assum}\\label{eqn-f}\n\t\tLet $I_d$ be the set defined by\n\t\t\\begin{equation}\\label{DEG-POL}\n\t\tI_d=\\begin{cases}\n\t\t\\mathbb{N}:=\\{1, 2, 3, \\ldots\\} \\text{ if } d=2,\\\\\n\t\t\\{1\\}, \\text{ if } d=3.\n\t\t\\end{cases}\n\t\t\\end{equation}\n\t\tWe fix $N\\in I_d\n\t\t$ and $a_k\\in \\mathbb{R}$, $k=0,\\ldots, N$,\n\t\twith $a_N<0$. We define a function $\\tilde{f}:[0,\\infty) \\rightarrow \\mathbb{R}$ by\n\t\t$$ \\tilde{f}(r)=\\sum_{k=0}^N a_k r^k, \\text{ for all } r\\in \\mathbb{R}_+.$$\n\t\tWe define a map $f:\\mathbb{R}^3\\rightarrow \\mathbb{R}^3$ by $f(\\d)=\\tilde{f}(\\vert \\d\\vert^2)\\d$ where $\\tilde{f}$ is as above.\n\t\t\n\t\tWe now assume that there exists $F: \\mathbb{R}^3 \\rightarrow \\mathbb{R}$ a Fr\\'echet differentiable map such that $$ F^\\prime(\\d)[\\mathbf{g}]= f(\\d)\\cdot \\mathbf{g},\n\t\\text{ $\\d\\in \\mathbb{R}^d$, $\\mathbf{g}\\in \\mathbb{R}^d$}.$$ Note that if $\\tilde{F}$ is a map such that $\\tilde{F}^\\prime= \\tilde{f}$ and $\\tilde{F}(0)=0$, then, there is $U$ is a polynomial function with $\\text{deg}(U)\\le N$ and $a_{N+1}<0$ such that \n\t$ \\tilde{F}(r)=a_{N+1}r^{N+1}+U(r).$\n\t\\end{assum}\n\n\t\n\t\\begin{Rem}\\label{REM-H2}\n\t\t\\begin{enumerate}[(i)]\n\t\t\\item \\label{Itemi:REM-H1 } There exists a constant $\\ell_3>0$ such that\n\t\t\t\\begin{align}\n\t\t\t\\lvert \\tilde{f}^{\\prime\\prime}(r) \\rvert\\le \\ell_3(1+r^{N-2}), \\,\\, r>0.\\label{ST6-B-2}\n\t\t\t\\end{align}\n\t\t\\item \\label{Itemi:REM-H2} From \\eqref{ST6-B-2}, we infer that there exist $c_0,c_1,c_3>0$ such that\n\t\t\t$$ \\lvert f(\\mathbf{n})\\rvert \\le c_0 (1+\\vert \\d\\vert^{2N+1}), \\text{ } \\lvert f^{\\prime}(\\d)\\rvert\\le\n\t\t\tc_1(1+\\vert \\mathbf{n} \\vert^{2N}) \\text{ and } \\lvert f^{\\prime \\prime}(\\d)\\rvert\\le c_2 (1+\\vert \\mathbf{n} \\vert^{2N-1}) \\text{ for all } \\mathbf{n}\\in \\mathbb{R}^n.$$\n\t\t\t\\item Let $\\tilde{q}=4N+2$. It is easy to show that there exists $C>0$ such that \\mbox{ for all } $\\d\\in \\mathrm{H}^2$\n\t\t\t\\begin{align}\n\t\t\t\\lVert \\mathrm{A}_1 \\d\\rVert^2=&\\lVert \\mathrm{A}_1 \\d +f(\\d)-f(\\d)\\rVert^2\n\t\t\t\\le 2 \\lVert \\mathrm{A}_1 \\d -f(\\d)\\rVert^2+2 \\lVert f(\\d)\\rVert^2,\\nonumber \\\\\n\t\t\t\\le & 2 \\lVert \\mathrm{A}_1 \\d -f(\\d)\\rVert^2+C \\lVert \\d\\rVert^{\\tilde{q}}_{\\mathbf{L}^{\\tilde{q}}}+C.\\label{bigdandel}\n\t\t\t\\end{align}\n\t\n\t\t\t\n\t\t\t\\item Since the norm $\\lVert \\cdot \\rVert_2$ is equivalent to $\\lVert \\cdot \\rVert + \\lVert \\mathrm{A}_1 \\cdot \\rVert $ on $D(\\mathrm{A}_1)$, there exists $C>0$\n\t\t\tsuch that\n\t\t\t\\begin{equation}\\label{bigdanh2}\n\t\t\t\\lVert \\d\\rVert^2_2\\le C (\\lVert \\mathrm{A}_1 \\d -f(\\d)\\rVert^2\n\t\t\t+\\lVert \\d\\rVert^{\\tilde{q}}_{\\mathbf{L}^{\\tilde{q}}}+1), \\mbox{ for all }\\d\\in D(\\mathrm{A}_1).\n\t\t\t\\end{equation}\n\t\t\t\n\t\t\t\\item\\label{Rem-Linf-H1Delta}\n\t\t\tSince $\\mathbf{H}^1\\hookrightarrow \\mathbf{L}^{4N+2}$, $N \\in I_d$, we infer from \\eqref{bigdanh2} that $\\mathbf{n}\\in \\mathbf{H}^2$ if $\\mathbf{n} \\in \\mathbf{H}^1$ and $\\mathrm{A}_1 \\mathbf{n} -f(\\mathbf{n})\\in \\mathbf{L}^2$.\n\t\t\\end{enumerate}\n\t\\end{Rem}\n\t\t\\begin{Rem}\\label{Rem:Ginz-Land-General}\n\t\tLet $\\varepsilon>0$ and $\\tilde{f}_\\varepsilon(r):=\\frac{1}{\\varepsilon^2}(-r+1)$, $r\\in [0,\\infty) $. Examples of maps $f$ and $F$ satisfying Assumption \\ref{eqn-f} are the following\n\t\t$$f(\\d):=\\tilde{f}(\\vert \\d \\vert^2)\\d=\\frac{1}{\\varepsilon^2}(1-\\lvert \\d \\rvert^2)\\d \\text{ and } F(\\d):=\\frac1{4\\varepsilon^2} [\\tilde{f}(\\vert \\d \\vert^2) ]^2, \\; \\d\\in \\mathbb{R}^d.$$\n\t\\end{Rem}\n\t\\subsection{The assumption on the coefficients of the noise}\n\t\n\t\\begin{assum}\\label{Assum:Usual hypotheses}\n\t\tThroughout this paper we are given a complete filtered probability space $(\\Omega,\n\t\t\\mathcal{F}, \\mathbb{P})$\n\t\twith the filtration $\\mathbb{F}=\\{\\mathcal{F}_t: t\\geq 0\\}$\n\t\tsatisfying the usual hypothesis, \\textit{i.e.},\n\t\tthe filtration is right-continuous and all null sets of $\\mathcal{F}$ are elements of $\\mathcal{F}_0$.\n\t\\end{assum}\n\tThroughout, let $\\mathrm{K}_1$ be a separable Hilbert space, and $W_1=(W_1(t))_{t\\geq 0}$ and \n\t$W_2=(W_2(t))_{t\\geq 0}$ be independent $\\mathrm{K}_1$-cylindrical\n\tWiener process and standard Brownian motion on $(\\Omega,\n\t\\mathcal{F},\\mathbb{F}, \\mathbb{P})$. If $\\mathrm{K}=\\mathrm{K}_1\\times \\mathbb{R}$ then we can assume that $W=(W_1(t),W_2(t))$ is $\\mathrm{K}$-cylindrical\n\tWiener process. \n\t\n\n\tLet $\\tilde{\\mathrm{K}}$ and $\\tilde{\\mathrm{H}}$ be a separable Hilbert and Banach spaces. We denote by $\\gamma(\\tilde{\\mathrm{K}}, \\tilde{\\mathrm{ H}})$ the space of $\\gamma$-radonifying operators which generalises the space of Hilbert-Schmidt operators $\\mathcal{T}_2(\\tilde{\\mathrm{K}}, \\tilde{\\mathrm{ H}})$ if $\\tilde{\\mathrm{ H}}$ is a separable Hilbert space, see \\cite{ZB-97}. Let $\\mathscr{M}^2(\\Omega\\times [0,T]; \\mathcal{T}_2(\\tilde{\\mathrm{K}}, \\tilde{\\mathrm{H}} ))$ the space of all equivalence classes of progressively measurable processes $\\Psi: \\Omega\\times [0,T]\\to\\mathcal{T}_2(\\tilde{\\mathrm{K}}, \\tilde{\\mathrm{H}} )$ satisfying\n\t$$ \\mathbb{E}\\int_0^T \\Vert \\Psi(s)\\Vert^2_{\\mathcal{T}_2(\\tilde{\\mathrm{K}}, \\tilde{\\mathrm{H}} )}ds <\\infty.$$\n\t\tFor a $\\tilde{\\mathrm{K}}$-cylindrical Wiener process $\\tilde{W}$ and $\\Psi \\in\\mathscr{M}^2(\\Omega\\times [0,T]; \\mathcal{T}_2(\\tilde{\\mathrm{K}}, \\tilde{\\mathrm{H}} ))$ the process $M$ defined by\n\t$ M(t) =\\int_0^t \\Psi(s)d\\tilde{W}(s), t\\in [0,T],$ is a $\\tilde{\\mathrm{H}}$-valued martingale. \n\tFor more detail on the theory of stochastic integration we refer to \\cite[Section 26 ]{Metivier_1982} and \\cite[Chapter 4]{DP+JZ-14}. \n\nLet $G$ be the map defined in \\eqref{Eq:LinG} and $G^2=G\\circ G$. Then, we have the following relation identity\n\tStratonovich and It\\^o's integrals, see \\cite{Brz+Elw_2000},\n\t\\begin{equation*}\n\tG(\\mathbf{n})\\circ dW_2= \\frac 12 G^2(\\mathbf{n}) \\,dt +\n\tG(\\mathbf{n})\\,dW_2.\n\t\\end{equation*}\n\n\t\n\t\n\t\n\tWe now introduce the standing set of hypotheses on the function $S$.\n\t\n\t\\begin{assum}\\label{HYPO-ST}\n\t\tWe assume that $S: \\mathrm{H}\\to \\mathcal{T}_2(\\mathrm{K}_1,\\mathrm{V})$ is a globally Lipschitz map. In particular,\n\t\tthere exists $\\ell_5\\geq 0$\n\t\tsuch that\n\t\t\\begin{equation}\\label{Eq:Hypo-ST}\n\t\t\\lVert S(\\mathbf{u})\\rVert^2_{\\mathcal{T}_2(\\mathrm{K}_1,\\mathrm{V})}\\leq \\ell_5 (1+\\lvert \\mathbf{u} \\rvert_{\\mathrm{L}^2}^2),\\;\\; \\mbox{ for all } \\mathbf{u} \\in \\mathrm{H}.\n\t\t\\end{equation}\n\t\\end{assum}\n\t\\begin{Rem}\\label{Rem:HYPO-ST}\n\t\tNotice that the assumption \\eqref{Eq:Hypo-ST} implies that there exists a constant $\\ell_6>0$ such that\n\t\t\\begin{equation}\\label{Eq:Hypo-ST-Rem}\n\t\t\\lVert S(\\mathbf{u})\\rVert^2_{\\mathcal{T}_2(\\mathrm{K}_1,\\mathrm{V})}\\leq \\ell_5 (1+\\lvert \\nabla \\mathbf{u} \\rvert_{\\mathrm{L}^2}^2),\\;\\; \\mbox{ for all } \\mathbf{u} \\in \\mathrm{H}.\n\t\t\\end{equation}\n\t\\end{Rem}\n\n\n\n\t\n\t\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t%\n\n\n\n\n\n\n\n\t%\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\\section{Existence and uniqueness of local and global strong Solution}\\label{SLC-Sect4}\nUsing the notations of Section \\ref{sec-spaces-2}, the system \\eqref{eqn-SLQE-v}-\\eqref{Init-Cond} can be written in the abstract\n\tform\n\t\\begin{align}\n\t&d\\mathbf{v}(t)+\\biggl(\\mathrm{A}\\mathbf{v}(t)+B(\\mathbf{v}(t),\n\t\\mathbf{v}(t))+M(\\mathbf{n}(t))\\biggr)dt=S(\\mathbf{v}(t))dW_1,\\label{ABS-v1}\\\\\n\t&d\\mathbf{n}(t)+\\biggl(\\mathrm{A}_1\\mathbf{n}(t)+ \\tilde{B}(\\mathbf{v}(t),\\mathbf{n}(t))-\n\tf(\\mathbf{n}(t))-\\frac 12 G^2(\\mathbf{n}(t))\\biggr)dt=G(\\mathbf{n}(t))dW_2,\\\\\n\t\\label{ABt-d1}\n\t& \\v(0)=v_0\n\t\\text{ and } \\d(0)=d_0.\n\t\\end{align}\nIn this section we prove the existence and uniqueness of the strong solution to problem \\eqref{ABS-v1}-\\eqref{ABt-d1}. \n\n\t\\subsection{Definition of local solutions}\n\tLet $(B_i, \\lVert \\cdot \\rVert_{B_i})$, $i=1, 2$, be two Banach spaces. We endow $B_1\\times B_2$ with the norm $\n\t\\rVert(b_1,b_2)\\lVert=\\sqrt{\\lVert b_1\\rVert_{B_1}^2 +\\lVert b_2\\rVert_{{B}_2}^2}.$\n\tHenceforth, we put\n\t\\begin{equation}\\label{eqn-spaces}\n\t\\mathscr{H}=\\mathrm{H}\\times \\mathbf{X}_{\\frac12}, \\;\\; \\mathscr{V}=\\mathrm{V}\\times \\mathbf{X}_{1} \\mbox{ and } \\mathscr{E}=\\mathrm{V}_1 \\times \\mathbf{X}_{\\frac32}.\n\t\\end{equation}\n\tNext, we denote by $\\{\\mathbb{S}_1(t)\\}_{t\\geq 0}$ and $\\{\\mathbb{T}(t)\\}_{t\\geq 0}$ the analytic semigroups \n\tgenerated by\n\t$-\\mathrm{A}$ on $\\mathrm{H}$ and by $\\mathrm{A}_1$ on $\\mathbf{L}^2$, respectively. \nIt is well-known that \n\tthe space $\\mathbf{X}_{\\frac12}$ is invariant wrt $\\{\\mathbb{T}(t)\\}_{t\\geq 0}$. The restriction of $\\{\\mathbb{T}(t)\\}_{t\\geq 0}$ to $\\mathbf{X}_{\n\t\t\\frac12}$ is also an analytic semigroup which will be denoted by $\\{\\mathbb{S}_2(t)\\}_{t\\geq 0}$.\n\tThe minus infinitesimal\n\tgenerator $\\tilde{\\mathrm{A}}_1$ of $\\{\\mathbb{S}_2(t)\\}_{t\\geq 0}$ is the part\n\tof $\\mathrm{A}_1$ on $\\mathbf{X}_{\\frac12}$, that is,\n\t\\begin{align*}\n\tD(\\tilde{\\mathrm{A}}_1)=\\{\\mathbf{u} \\in D(\\mathrm{A}_1): \\mathrm{A}_1\\mathbf{u}\\in \\mathbf{X}_{\\frac12}\\},\\;\\;\n\t\\tilde{\\mathrm{A}}_1\\mathbf{u}=\\mathrm{A}_1\\mathbf{u} \\text{ for all } \\mathbf{u} \\in D(\\tilde{\\mathrm{A}}_1).\n\t\\end{align*}\n\tNote that $\\mathbf{X}_{\\frac32}\\subset D(\\tilde{\\mathrm{A}}_1).$\n\tWith all the above notation, the problem \\eqref{ABS-v1}-\\eqref{ABt-d1} can be rewritten as the following stochastic evolution equation in the space $\\mathscr{H}$,\n\t\\begin{equation}\\label{ABSTRACT-LC}\n\td\\mathbf{y}(t) +\\mathbf{A}\\mathbf{y}(t) dt+\\mathbf{F}(\\mathbf{y}(t))\n\tdt+\\mathbf{L}(\\mathbf{y}(t))dt=\\mathbf{G}(\\mathbf{y}(t)) d{W}(t),\n\t\\end{equation}\n\twhere, for $\\mathbf{y}=(\\mathbf{v}, \\d)\\in E$ and $k=(k_1,k_2)\\in \\mathrm{K}$,\n\t\n\t\\begin{equation}\\label{eqn-def-A-F}\n\t\\mathbf{A}\\mathbf{y}=\\begin{pmatrix} \\mathrm{A} \\mathbf{v} \\\\\n\t \\mathrm{A}_1 \\mathbf{n}\n\t\\end{pmatrix},\\;\\;\n\t\\mathbf{F}(\\mathbf{y})=\\begin{pmatrix} B(\\mathbf{v},\\mathbf{v})+M(\\mathbf{n})\\\\\n\t\\tilde{B}(\\mathbf{v},\\mathbf{n})-f(\\mathbf{n})\n\t\\end{pmatrix},\n\t\\end{equation}\n\t\\begin{equation}\\label{eqn-def-L-G}\n\t\\mathbf{L}(\\mathbf{y})=\\begin{pmatrix} 0\\\\ -\\frac 1 2 G^2(\\mathbf{n})\n\t\\end{pmatrix}, \\mathbf{G}(\\mathbf{y})k=\\begin{pmatrix}S(\\mathbf{u})k_1\\\\\n\tG(\\mathbf{n})k_2 \\end{pmatrix}.\n\t\\end{equation}\n\tThe operator $-\\mathbf{A}$ generates an analytic semigroup $\\{\\mathbb{S}(t)\\}_{t\\geq 0}$ on $\\mathscr{H}=\\mathrm{H}\\times \\mathbf{X}_{\\frac12}$\n\tdefined by\n\t\\begin{equation*}\n\t\\mathbb{S}(t)\\begin{pmatrix}\\mathbf{v}\\\\\\d\n\t\\end{pmatrix}=\\begin{pmatrix}\\mathbb{S}_1(t)\\mathbf{v}\\\\\n\t\\mathbb{S}_2(t)\\d\n\t\\end{pmatrix}, \\;\\; (\\mathbf{v},\\d)\\in \\mathscr{H}.\n\t\\end{equation*}\n\n\n\tImportant properties of $\\{\\mathbb{S}(t): t\\ge 0\\}$ are given in the next two lemma.\n\t\\begin{lem}\\label{SEM-1} \\label{SEM-2}\n\t\tLet $T\\in (0,\\infty)$, $\\mathbf{g}=\\begin{pmatrix}\n\t\t\\tilde{g} \\\\ g\n\t\t\\end{pmatrix}\\in L^2(0,T; \\mathrm{H} \\times \\mathbf{X}_{\\frac12})$ and\n\t\t$\n\\begin{pmatrix}\n\t\\mathbf{v}(t)\\\\\n\t\\mathbf{n}(t)\n\t\\end{pmatrix}= \\int_0^t\\mathbb{S}(t-s) \\mathbf{g}(s)\\, ds, \\;\\; t\\ge 0.$ Then, there exists $c_1>0$ such that\n\t\t\\begin{equation*}\n\t\t\\left \\lVert \\begin{pmatrix}\n\t\t\\mathbf{v}\\\\\\mathbf{n}\n\t\t\\end{pmatrix} \\right\\rVert_{C([0,T];\\mathrm{V} \\times \\mathbf{X}_{1})}+\\left \\lVert \\begin{pmatrix}\n\t\t\\mathbf{v}\\\\\\mathbf{n}\n\t\t\\end{pmatrix} \\right\\rVert_{L^2(0,T;D(\\mathrm{A}) \\times \\mathbf{X}_{\\frac32}) }\\le c_1 \\left \\lVert \\begin{pmatrix}\n\t\t\\tilde{g} \\\\ g\n\t\t\\end{pmatrix}\\right \\rVert_{L^2(0,T;\n\t\t\t\\mathrm{H}\\times \\mathbf{X}_{\\frac12})}.\n\t\t\\end{equation*}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tThis result is well-known and is a special case of \\cite[Lemma 1.2]{Pardoux}.\n\t\\end{proof}\n\t\\begin{lem}\\label{SEM-3}\n\t\t\t\tLet $T\\in (0,\\infty)$, $\\zeta= \\begin{pmatrix}\n\t\t\t\t\\zeta_1\\\\ \\zeta_2\n\t\t\t\t\\end{pmatrix}\\in \\mathscr{M}^2(0,T;\\mathrm{H}\\times \\mathbf{X}_1) $ and \n\t\t\t\t$ \\begin{pmatrix}\n\t\t\t\t\\mathbf{w}_1(t)\\\\ \\mathbf{w}_2(t)\n\t\t\t\t\\end{pmatrix} = \\int_0^t \\mathbb{S}(t-s) \\zeta(s) dW(s), t \\ge 0. $ Then, \n\t\tthere exists $C>0$ such that \n\t\t\\begin{equation*} \\mathbb{E} \\left \\lVert\n\t\\begin{pmatrix}\n\t\\mathbf{w}_1 \\\\ \\mathbf{w}_2\n\t\\end{pmatrix} \\right \\rVert^2_{C([0,T];\\mathrm{V} \\times \\mathbf{X}_1 )}+\\mathbb{E}\\left \\lVert\n\t\t\\begin{pmatrix}\n\t\t\\mathbf{w}_1 \\\\ \\mathbf{w}_2\n\t\t\\end{pmatrix} \\right \\rVert^2_{L^2(0,T;D(\\mathrm{A}) \\times \\mathbf{X}_{\\frac32})}\\le C \\mathbb{E}\\left \\lVert\n\t\\begin{pmatrix}\n\t\\zeta_1\\\\ \\zeta_2\n\t\\end{pmatrix}\\right \\rVert^2_{L^2(0,T;\\mathrm{V} \\times \\mathbf{X}_1)},\\;\\; T\\ge 0.\n\t\t\\end{equation*}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tThis result is also well-known, see \\cite[Lemma 1.4]{Pardoux}.\n\t\\end{proof}\n\n\tLet us recall the following notations\/definition which are borrowed\n\tfrom \\cite{Kunita-90}, see also \\cite{Brz+Elw_2000}.\n\n\n\n\t\\begin{Def}\\label{def-accessible stopping time}\n A random function $\\tau:\\Omega \\to [0,\\infty]$ is called a stopping time, see \\cite[Definition I.2.1]{Kar-Shr-96}, \\cite[Definition 4.1]{Metivier_1982} and \\cite[section III.5]{Elw_1982}, iff for each $t\\geq 0$, the set\n$\\{\\omega \\in \\Omega: t< \\tau(\\omega)\\} \\in \\mathcal{F}_t$ (or equivalently, $\\{\\omega \\in \\Omega: \\tau(\\omega) \\leq t\\} \\in \\mathcal{F}_t$). A stopping time\n\t\t$\\tau:\\Omega \\to [0,\\infty]$ is called accessible, see \\cite[section 2.1, p. 45]{Kunita-90}, iff there exists an increasing\n\t\tsequence\\footnote{In the sense that for all $n\\in \\mathbb{N}$, $\\tau_n \\leq\n\t\t\\tau_{n+1}$, $\\mathbb{P}$-a.s. } of stopping times $\\tau_n{:\\Omega \\to [0,\\infty)}$ such that $\\mathbb{P}$-a.s. \n\t\\begin{inparaenum}\n\t\t\\item[(i)] for all $n\\in \\mathbb{N}$, $\\tau_n <\n\t\t\\tau$;\n\t\t\\item[(ii)] and $\\lim_{n\\to \\infty} \\tau_n =\\tau$.\n\t\\end{inparaenum}\n\nThe sequence $(\\tau_n)_{n\\in \\mathbb{N}}$ as above is usually called an announcing sequence for $\\tau$.\n\t\\end{Def}\n\n\\begin{Rem}\\label{rem-predictable stopping time}\nUnder the Assumption \\ref{Assum:Usual hypotheses} we have the following facts.\n\\begin{trivlist}\n\n\t\t\\item[(i)]\nIt follows from \\cite[Proposition 6.6 (3)]{Metivier_1982} that a stopping time is accessible if and only if it is predictable. Let us recall, see \\cite[Definition 4.9]{Metivier_1982}, that\na stopping time $\\tau$ is predictable iff its graph $[\\tau]:=\\{(t,\\omega)\\in [0,\\infty)\\times\\Omega: t=\\tau(\\omega)\\}$ is a predictable set.\nThe $\\sigma$-field $\\mathcal{P}$ of predictable sets is generated by the family $\\mathcal{R}:=\\{ (s,t]\\times F: 0\\leq s\\leq t, F\\in \\mathcal{F}_s\\}\\cup \\{\\{0\\} \\times F: F\\in \\mathcal{F}_0\\}$, see \\cite[Theorem 3.3]{Metivier_1982}.\n\t\t\\item[(ii)] If $(\\tau_n)_{n \\in \\mathbb{N}}$ is a sequence of accessible stopping times, then $\\sup_{n\\in \\mathbb{N}} \\tau_n$ is also an accessible stopping times, see \\cite[Proposition 6.6]{Metivier_1982}. In particular, If $\\tau$ and $\\sigma$ accessible stopping times with announcing sequences $(\\tau_n)_{n \\in \\mathbb{N}}$ and $(\\sigma_n)_{n \\in \\mathbb{N}}$, then $\\tau\\vee \\sigma$ is an accessible stopping time with announcing sequence $(\\tau_n \\vee \\sigma_n)_{n \\in \\mathbb{N}}$. Furthermore, if $\\mathcal{A}$ is an arbitrary family of accessible stopping times then a family\n\t\t\\[\n\t\t\\mathcal{B}:=\\{ \\sup \\mathcal{C}: \\mathcal{C} \\mbox{ is a finite subset of } \\mathcal{A} \\}\n\t\t\\]\n\t\tis also a family of accessible stopping times such that $\\mathcal{A} \\subset \\mathcal{B}$ and the supremum of each finite subset of $\\mathcal{B}$ belongs to $\\mathcal{B}$. In particular, if $\\Delta$\n\t\tis the family of all accessible stopping times, then the supremum of each finite subset of $\\Delta$ belongs to $\\Delta$.\n\\end{trivlist}\n\n\\end{Rem}\n\n\t\\textbf{Notation}. For a stopping time $\\tau$ we set\n\n\t\\[ \\Omega_t(\\tau) \n\t\\{ \\omega \\in \\Omega : t < \\tau(\\omega)\\},\n\t\\\n\t\\[\n\t[0,\\tau)\\times \\Omega \n\t\\{ (t,\\omega) \\in [0,\\infty)\\times \\Omega: 0\\le t < \\tau(\\omega)\n\t\\}.\n\t\\\n\t\n\t\\begin{Def}\\label{def-adapted process}\n Assume that $X$ is a {topological space}.\n\t\tAn $X$-valued process {local} process $\\eta : [0,\\tau) \\times \\Omega \\to X$ (we will also write $ \\eta(t)$, $t< \\tau$) is\n\t\tadmissible iff\n\t\t\\begin{inparaenum}\n\t\t\t\\item[(i) ] it is adapted, i.e. $\\eta|_{\\Omega_t(\\tau)}: \\Omega_t(\\tau) \\to\n\t\t\tX$ is $\\mathcal{F}_t$ measurable, for all $t\\ge 0$; \\item[(ii)]\n\t\t\tfor almost all $\\omega \\in \\Omega$, the function $[0,\n\t\t\t\\tau(\\omega))\\ni t \\mapsto \\eta(t, \\omega) \\in X$ is continuous.\n\t\t\t\\end{inparaenum}\n\t\t%\n\t\t\n\t\tTwo {local} processes $\\eta_i: [0,\\tau_i) \\times \\Omega \\to X$,\n\t\t$i=1,2$ are called equivalent (and we will write $(\\eta_1,\\tau_1) \\sim\n\t\t(\\eta_2,\\tau_2)$) iff $\\tau_1=\\tau_2$ $\\mathbb{P}$-a.s. and, for all $t>0$,\n\t\tthe following condition holds\n\t\t\\[\n\t\t\\eta_1(\\cdot,\\omega)= \\eta_2(\\cdot,\\omega) \\mbox{ on } [0,t], ; \\mbox{ \t\tfor a.e. $\\omega \\in \\Omega_t(\\tau_1)\\cap \\Omega_t(\\tau_2)$}.\n\t\t\\]\n\t\tNote that if { two local} admissible processes $\\eta_i : [0,\\tau_i) \\times \\Omega \\to\n\t\tX$, $i=1,2$ are such that for all $t>0$\n\t\t$\\eta_1(t)|_{\\Omega_t(\\tau_1)}= \\eta_2(t)|_{\\Omega_t(\\tau_2)}$\n\t\t$\\mathbb{P}$-a.s., then they are equivalent.\n\t\\end{Def}\n\n\\begin{Rem}\\label{rem-adapted local process}\nLet $\\tau$ be an accessible stopping time with an announcing sequence $(\\tau_n)_{n \\in \\mathbb{N}}$ and $\\eta : [0,\\tau) \\times \\Omega \\to X$ is a local process.\n Kunita \\cite[section 2.1, p. 46]{Kunita-90} defined $\\eta$ to be a local adapted process iff\n the following condition is satisfied for every $n$, the stopped process $(\\eta_{t\\wedge \\tau_n})_{t\\geq 0}$ is adapted. We do not know how condition (i) from Definition \\ref{def-adapted process} is related to Kunita's definition. \n\\end{Rem}\n\t\\begin{Def}\\label{def-progrssively measurable process}\nLet $\\tau$ be an accessible stopping time with an announcing sequence $(\\tau_n)_{n \\in \\mathbb{N}}$. Motivated by \\cite[Proposition 2.18]{Kar-Shr-96}, a local process $\\eta : [0,\\tau) \\times \\Omega \\to X$ is called progressively\nmeasurable iff for every $n$, the stopped process $(\\eta_{t\\wedge \\tau_n})_{t\\geq 0}$ is progressively measurable. \n\\end{Def}\n\tWe now define some concepts of solution to \\eqref{ABSTRACT-LC}, see \\cite[Definition 4.2]{Brz+Millet_2012} or \\cite[Definition\n\t2.1]{Mikulevicius}.\n\t\\begin{Def}\\label{def-local solution} Let $\\mathbf{y}_0:\\Omega \\to V$ be $\\mathcal{F}_0$-measurable random variable satisfying $\\mathbb{E} \\Vert \\mathbf{y}_0\\Vert^2_{\\mathscr{V}}<\\infty$. A local\n\t\tsolution to problem \\eqref{ABSTRACT-LC}(with the initial time\n\t\t$0$) is a pair $(\\mathbf{y},\\tau)$ such that\n\t\t\\begin{enumerate}[(1)]\n\t\t\t\\item $\\tau$ is an accessible stopping time with an announcing sequence $(\\tau_n)_{n\\in \\mathbb{N}}$,\n\t\t\t\\item $\\mathbf{y}: [0,\\tau)\\times \\Omega \\to V$ is an admissible process,\n\t\t\t\\item for every $n\\in \\mathbb{N}$ and $t\\in [0,\\infty)$,\n\t\t\twe have\n\t\t\t\\begin{align}\n\t\t\t\\mathbb{E}\\Big( \\sup_{s\\in [0,t\\wedge \\tau_n]} \\Vert\n\t\t\t\t\\mathbf{y}(s)\\Vert^2_{\\mathscr{V}} +\\int_0^{t\\wedge \\tau_n} \\Vert \\mathbf{y}(s)\\Vert_\\mathscr{E}^2 \\,\n\t\t\t\tds\\Big)<\\infty,\\label{eq-locsol_01-a}\n\t\t\t\t\\end{align}\n\t\t\t\tand $\\mathbb{P}$-a.s.\n\t\t\t\t\\begin{align}\n\t\t\t\t\\mathbf{y}(t\\wedge \\tau_n)= \\mathbb{S}(t\\wedge\n\t\t\t\t\\tau_n)\\mathbf{y}_0-\\int_0^{t\\wedge \\tau_n}\n\t\t\t\t\\mathbb{S}(t\\wedge\\tau_n-s)[ \\mathbf{F}(\\mathbf{y}(s\\cnew{\\wedge \\tau_n}))+\\mathbf{L}(\\mathbf{y}(s\\cnew{\\wedge \\tau_n})] \\,ds +I_{\\tau_n}(t\\wedge \\tau_n), \\label{eq-locsol_01-b}\n\t\t\t\\end{align}\nwhere $I_{\\tau_n}$ is a continuous $V$-valued process process defined by\n\t\t\t\\begin{equation}\\label{eq-locsol_01-c}\n\t\t\t\\begin{split}\nI_{\\tau_n}(t):= \\int_0^t\\mathds{1}_{[0,\\tau_n)}(s)\\mathbb{S}(t-s)\\mathbf{G}(\\mathbf{y}(s \\wedge \\tau_n))\\,\n\t\t\td{W}(s),\\;\\; t\\in[0,\\infty).\n\t\t\t\\end{split}\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\t\tAlong the lines of the paper \\cite{Brz+Elw_2000}, we say that a\n\t\tlocal solution $\\mathbf{y}(t)$, $t < \\tau$ is global iff\n\t\t$\\tau=\\infty$ $\\mathbb{P}$-a.s.\n\t\\end{Def}\nHereafter, we simply write local solution in place of local mild solution. \n\\begin{Rem}\\label{rem-stochastic interval}\n\\begin{trivlist}\n\\item[(i)] Since $\\tau_n$ is a stopping, the process $\\mathds{1}_{[0,\\tau_n)}(s)$, $s\\in [0,\\infty)$ is well-measurable, see \\cite[Proposition 4.2]{Metivier_1982}. Therefore, since by \\cite[Theorem 1.6]{Metivier_1982}, the $\\sigma$-field of well measurable sets is smaller than the $\\sigma$-field of progressively measurable sets, it follows\nthat the process $\\mathds{1}_{[0,\\tau_n)}(s)$, $s\\in [0,\\infty)$ is progressively measurable. In particular, the integrand in \\eqref{eq-locsol_01-c} is progressively measurable.\n\\item[(ii)] On the other hand, one could use in \\eqref{eq-locsol_01-c} a process $\\mathds{1}_{[0,\\tau_n]}(s)$, $s\\in [0,\\infty)$, which according \\cite[Proposition 4.4]{Metivier_1982}, is predictable. However, here we still stick to the processes as in (i).\n\\end{trivlist}\n\\end{Rem}\n\\begin{Rem}\\label{rem-local solution} Suppose that $\\tau:\\Omega \\to [0,\\infty)$ is an accessible stopping time and $\\mathbf{y}: [0,\\tau]\\times \\Omega \\to V$ is an admissible process such that\nfor every $t \\in [0,\\infty)$\n\t\t\t\\begin{align}\n\t\t&\t\\mathbb{E}\\Big( \\sup_{s\\in [0,t\\wedge \\tau]} \\Vert\n\t\t\t\t\\mathbf{y}(s)\\Vert^2_{\\mathscr{V}} +\\int_0^{t\\wedge \\tau} \\Vert \\mathbf{y}(s)\\Vert_\\mathscr{E}^2 \\,\n\t\t\t\tds\\Big)<\\infty,\\label{eq-locsol_01-a2}\\\\ \n\t&\t\t\\mathbf{y}(t\\wedge \\tau)= \\mathbb{S}(t\\wedge\n\t\t\t\\tau)\\mathbf{y}_0-\\int_0^{t\\wedge \\tau}\n\t\t\t\\mathbb{S}(t\\wedge\\tau-s)[ \\mathbf{F}(\\mathbf{y}(s\\wedge \\tau))+\\mathbf{L}(\\mathbf{y}(s \\wedge \\tau)]\\, ds\n +I_{\\tau}(t\\wedge \\tau), \\;\\;\\;\\mbox{ $\\mathbb{P}$-a.s.,} \\label{eq-locsol_01-d2}\n\t\t\t\\end{align}\nwhere $I_{\\tau}$ is a continuous $V$-valued process process defined by\n\t\t\t\\begin{equation}\\label{eq-locsol_01-d}\n\t\t\t\\begin{split}\nI_{\\tau}(t):= \\int_0^t\\mathds{1}_{[0,\\tau)}(s)\\mathbb{S}(t-s)\\mathbf{G}(\\mathbf{y}(s \\wedge \\tau))\\,\n\t\t\td{W}(s),\\;\\; t\\in [0,\\infty).\n\t\t\t\\end{split}\n\t\t\t\\end{equation}\nLet us choose an announcing sequence $(\\tau_n)_{n\\in \\mathbb{N}}$ for $\\tau$. Then, by using \\cite[Lemma A.1]{Brz+Masl+Seidler_2005}, we can show that for every $n$, the conditions\n\\eqref{eq-locsol_01-a} and \\eqref{eq-locsol_01-b} are satisfied with $I_{\\tau_n}$ defined by \\eqref{eq-locsol_01-c}. Therefore we infer that the restriction of the process\n$\\mathbf{y}$ to the open stochastic interval $[0,\\tau)\\times \\Omega$ is a local solution to problem \\eqref{ABSTRACT-LC}.\n\\end{Rem}\n\t\n\tWe now introduce the definition of a maximal local solution.\n\t\n\t\\begin{Def}\\label{Def-maxsol-0}\n\t\tConsider a family $\\mathcal{ LS}$ of all local solution\n\t\t$(u,\\tau)$ to the problem \\eqref{ABSTRACT-LC}. For two\n\t\telements $(u,\\tau), (v,\\sigma) \\in \\mathcal{ LS} $ we write that\n\t\t$(u,\\tau)\\preceq (v,\\sigma)$ iff $\\tau \\leq \\sigma$ $\\mathbb{P}$-a.s. and\n\t\t$v_{\\vert [0,\\tau)\\times \\Omega} \\sim u$. Note that if\n\t\t$(u,\\tau)\\preceq (v,\\sigma)$ and $(v,\\sigma)\\preceq (u,\\tau)$,\n\t\tthen $(u,\\tau)\\sim (v,\\sigma)$. We write $(u,\\tau)\\prec\n\t\t(v,\\sigma)$ iff $(u,\\tau)\\preceq (v,\\sigma)$ and $(u,\\tau)\\not\\sim\n\t\t(v,\\sigma)$. Then, the pair $(\\mathcal{ LS},\\preceq)$ is \n\t\tpartially ordered. \n\t\tEach maximal element $(u,\\tau)$ in the set $(\\mathcal{\n\t\t\tLS},\\preceq)$\n\t\tis called a maximal local solution to the problem \\eqref{ABSTRACT-LC}. The existence of an upper bound of every non-empty chain of $(\\mathcal{\n\t\t\tLS},\\preceq)$ is justified by Amalgamation\n\t\tLemma \\ref{lem-amalgamation}. \\\\\n\t\tIf $(u, \\tau)$ is a maximal local solution to equation\n\t\\eqref{ABSTRACT-LC}, the stopping time $\\tau$ is called its\n\t\tlifetime.\n\t\\end{Def}\n\t\\subsection{Existence and uniqueness of a maximal local solution: 2D and 3D cases}\n\tBy using Theorem \\ref{thm_local} we will prove in this subsection that the problem \\eqref{ABSTRACT-LC} has a unique maximal local solution. In order to do this, we need to establish several auxiliary results. Throughout this subsection $d=2,3$ and $a=\\frac{d}4$.\n\t\\begin{lem}\\label{Local-LIP-Lem}\n\t\tThere exists $c_2>0$ such that for\n\t\tall $\\mathbf{n}_i\\in \\mathbf{H}^3$, i$=1,2$,\n\t\t\\begin{equation}\\label{local-Lip-F-2}\n\t\t\\begin{split}\n\t\t\\lvert M(\\mathbf{n}_1)-M(\\mathbf{n}_2)\\rvert_{\\mathrm{L}^2} \\le c_2\\biggl(\\lVert \\d_1-\\d_2\\rVert_2\n\t\t\\lVert \\d_1\\rVert_2^{1-a} \\lVert \\d_1\\rVert_3^a+ \\lVert\n\t\t\\d_1-\\d_2\\rVert^{1-a}_2 \\lVert \\d_1-\\d_2\\rVert_3^a \\lVert \\d_2\\rVert_2\n\t\t\\biggr).\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tThe proof of this result can be found in \\cite[Lemma 6.4]{BHP18}\n\t\\end{proof}\n\n\n\n\t\\begin{lem}\\label{Local-LIP-Lem-2}\n\tThere exist $c_3>0$ such that for all $(\\mathbf{v}_i,\n\t\t\\mathbf{n}_i)\\in \\mathscr{E}$, i$=1,2$,\n\t\t\\begin{equation}\\label{local-Lip-F-3}\n\t\t\\begin{split}\n\t\t\\lVert \\tilde{B}(\\mathbf{v}_1,\\mathbf{n}_1)-\\tilde{B}(\\mathbf{v}_2,\\mathbf{n}_2)\\rVert_1 \\le c_3\n\t\t\\biggl(\\lVert \\nabla(\\mathbf{v}_1-\\mathbf{v}_2)\\rVert \\lVert\n\t\t\\mathbf{n}_1\\rVert_2^{1-a}\\lVert \\d_1\\rVert_3^a\n\t\t+\\lVert(\\mathbf{n}_1-\\mathbf{n}_2)\\lVert^{1-a}_ 2 \\lVert(\\mathbf{n}_1-\\mathbf{n}_2)\\lVert^{a}_3\\lVert\n\t\t\\nabla\\mathbf{v}_2\\lVert\\biggr).\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tThroughout this proof $C>0$ is an universal constant which may change from one term to the other. \t\n\tLet $(\\mathbf{v}_i,\n\t\\mathbf{n}_i)\\in \\mathscr{E}$, i$=1,2$, and\n\t$(\\mathbf{w},\n\t\\bar{\\d})=(\\mathbf{v}_1-\\mathbf{v}_2, \\d_1-\\d_2). $\nSince $\\tilde{B}$ is bilinear, we have\n\t\\begin{equation}\n\t\\tilde{B}(\\mathbf{v}_1,\\mathbf{n}_1)-\\tilde{B}(\\mathbf{v}_2,\\mathbf{n}_2)=\\tilde{B}(\\mathbf{w},\\d_1)+\\tilde{B}(\\v_2, \\bar{\\d})=J_1 +J_2.\n\t\\end{equation}\n\tIn order to estimate $\\lVert J_i\\rVert_1$,$i=1,2,$, we only focus on estimating $\\lVert \\nabla J_i \\rVert$, because by \\eqref{EST-G1} estimating $\\lVert J_i\\rVert$\n\tis easy. By using the Leibniz rule, the H\\\"older inequality, \\eqref{GAG-l4} and\\eqref{GAG-LInf-2} we infer that\n\t\t\\begin{align*}\n\t\t\\lVert \\nabla J_1\\rVert\\le & C \\biggl(\\lVert \\nabla \\mathbf{w}\\rVert \\lVert \\nabla\n\t\t\\d_1\\rVert_{\\mathrm{L}^\\infty}+\\lVert \\mathbf{w}\\rVert_{\\mathrm{L}^4} \\lVert \\nabla^2\n\t\t\\d_1\\rVert_{\\mathrm{L}^4}\\biggr)\\\\\n\t\t\\le& C \\lVert \\mathbf{w} \\rVert_1 \\biggl(\\lVert \\nabla \\d_1\\rVert^{1-a}_1 \\lVert \\nabla \\d_1\\rVert^a_2 +\n\t\t\\lVert \\nabla^2 \\d_1\\rVert^{1-a}_{L^2} \\lVert \\nabla^2 \\d_1\\rVert_{1}^a \\biggr)\n\\le C \\lVert \\mathbf{v}_1-\\mathbf{v}_2 \\rVert_1 \\lVert \\d_1\\rVert^{1-a}_2 \\lVert \\d_1\\rVert^a_3.\n\t\t\\end{align*}\n\t\tIn a similar way, we can prove that\n\t\t\\begin{equation*}\n\t\t\\lVert \\nabla J_2\\rVert\\le C \\lvert \\nabla\n\t\t\\mathbf{v}_2 \\rvert_{\\mathrm{L}^2} \\lVert \\d_1-\\d_2\\rVert^{1-a}_2 \\lVert \\d_1-\\d_2\\rVert^a_3.\n\t\t\\end{equation*}\n\t\tThe inequality \\eqref{local-Lip-F-3} easily follows from \\eqref{EST-G1} and the estimates for \t$\\lVert \\nabla J_i\\rVert$, $i=1,2$, above.\n\t\\end{proof}\n\t\n\t\\begin{lem}\\label{Local-LIP-Lem-3}\n\t\tLet Assumption \\ref{eqn-f} be satisfied. Then, there exists $c_4>0$\n\t\tsuch that \n\t\t\\begin{equation}\\label{local-Lip-F-4}\n\t\t\\begin{split}\n\t\t\\lVert f(\\mathbf{n}_1)-f(\\mathbf{n}_2)\\rVert_1 \\le c_4\\Big[1+\\lVert \\mathbf{n}_1\\rVert_2^{2N}+\\lVert \\mathbf{n}_2\\rVert_2^{2N}\\Big]\\lVert \\mathbf{n}_1-\\mathbf{n}_2\\rVert_2, \\text{ for all $\\mathbf{n}_1, \\mathbf{n}_2\\in \\mathbf{X}_{1}\\cap \\mathbf{X}_{\\frac32}$}.\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\nLet\t$N \\in I_d$, $k\\in \\{0,\\ldots, N\\}$ and $f$ be as in Assumption \\ref{eqn-f}. By the Young inequality and the fact that $\\mathbf{H}^2 $ is an algebra, we infer that there exists $C>0$ such that for all $\\mathbf{n}_1,\\mathbf{n}_2\\in \\mathbf{H}^2$\n\t\t\\begin{equation}\\label{POL-EST-0}\n\t\t\\lVert\\lvert \\mathbf{n}_1 \\rvert^{2k} \\mathbf{n}_2\\rVert_2\\le C \\lVert \\mathbf{n}_1 \\rVert_2^{2k} \\lVert \\mathbf{n}_2\\rVert_2\\le C \\lVert \\mathbf{n}_2\\rVert_2(1+\\lVert \\mathbf{n}_1 \\lVert_2^{2N}.\n\t\t\\end{equation}\n\t\tThus, it is enough to establish \\eqref{local-Lip-F-4} for the leading term $a_N \\lvert \\mathbf{n} \\rvert^{2N}\\mathbf{n}$. For doing so, we have $$\\lvert \\mathbf{n}_1\\rvert^{2N}\\mathbf{n}_1-\\lvert \\mathbf{n}_2\\rvert^{2N}\\mathbf{n}_2= \\vert \\mathbf{n}_1\\vert^{2N}(\\mathbf{n}_1-\\mathbf{n}_2)+\n\t\t\\mathbf{n}_2(\\vert\\mathbf{n}_1\\vert-\\vert\\mathbf{n}_2\\vert)(\\sum_{k=0}^{2N-1}\\vert \\mathbf{n}_1\\vert^{2N-1-k}\\vert \\mathbf{n}_2\\rvert^{k} ),$$\n\t\tfrom which along with \\eqref{POL-EST-0} we infer that \\eqref{local-Lip-F-4} is true for the leading term $a_N \\lvert \\mathbf{n} \\rvert^{2N}\\mathbf{n}$.\n\t\\end{proof}\n\t\\begin{prop}\\label{SLC-ST}\n\tLet $\\alpha=\\frac d4$, $d=2,3$. If Assumption \\ref{eqn-f} is satisfied, then there exists $C_0>0$ such that for all $\\mathbf{y}_i=(\\mathbf{v}_i,\\d_i)$, \\,\\,\n\t\t$i=1,2$\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\lVert \\mathbf{F}(\\mathbf{y}_1) -\\mathbf{F}(\\mathbf{y}_2)\\rVert_{\\mathscr{H}} \\le C_0 \\Vert \\mathbf{y}_1-\\mathbf{y}_2\\Vert_{\\mathscr{V}}^{1-\\alpha}\n\t\t\\Big[\\Vert \\mathbf{y}_1-\\mathbf{y}_2\\Vert^{\\alpha}_{\\mathscr{V} } \\Vert \\mathbf{y}_1\\Vert^{1-\\alpha}_{\\mathscr{V}} \\Vert \\mathbf{y}_1\\Vert_{\\mathscr{E}}^\\alpha + \\Vert\n\t\t\\mathbf{y}_1-\\mathbf{y}_2\\Vert_{\\mathscr{E}}^\\alpha \\Vert\n\t\t\\mathbf{y}_2\\Vert_{\\mathscr{V}}\\Big]\\\\\n\t\t+c_4\\lVert \\mathbf{y}_1-\\mathbf{y}_2\\rVert_{\\mathscr{V} }\\Big[1+\\lVert \\mathbf{y}_1\\rVert_{\\mathscr{V}}^{2N}+\\lVert \\mathbf{y}_2\\rVert_{\\mathscr{V}}^{2N}\\Big].\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\\end{prop}\n\t\\begin{proof}\n\t\tThe proposition is an easy consequence of Lemmata \\ref{Local-LIP-Lem}-\\ref{Local-LIP-Lem-3}. Thus, we omit its proof.\n\t\\end{proof}\n\t\n\t\n\tUsing Theorems \\ref{Thm:LocalUniqueness}, \n\t\\ref{thm_local} and \\ref{thm_maximal-abstract} we obtain our first main result.\n\t\\begin{thm}\\label{LC-Local-Sol}\n\t\tLet $d=2,3$, $(\\v_0,\\d_0)\\in \\mathrm{L}^2(\\Omega;\\mathrm{V} \\times \\mathbf{X}_{1})$ be $\\mathcal{F}_0$-measurable, $\\mathbf{h}\\in \\mathbf{H}^2$. If Assumptions \\ref{eqn-f}, \\ref{HYPO-ST} and \\ref{Assum:Usual hypotheses} are satisfied, then the problem \\eqref{ABSTRACT-LC} has a unique maximal local solution $((\\mathbf{v};\\mathbf{n}), \\tilde{\\tau}_\\infty)$ satisfying the following properties.\n\t\t\\begin{enumerate}[(1)]\n\t\t\t\\item\\label{item-1} Given $R>0$ and $\\varepsilon >0$ there exists\n\t\t\t$\\tau(\\varepsilon,R)>0$ such that if $\\mathbb{E}\\Vert (\\v_0,\\d_0) \\Vert^{2}_{\\mathrm{V} \\times \\mathbf{X}_{1}} \\leq R^{2}$, then \n\t\t\t\\[{\\mathbb P}\\big(\\tilde{\\tau}_\\infty \\geq \\tau(\\varepsilon,R)\\big) \\geq\n\t\t\t1-\\varepsilon.\\]\n\t\t\t\\item \\label{THM-ii} We also have\n\t\t\t\\begin{align}\n\t\t\t&\\mathbb{P}\\left(\\{ \\tilde{\\tau}_\\infty <\\infty \\}\\cap \\{ \\lvert \\nabla \\mathbf{v}(t) \\rvert_{\\mathrm{L}^2} + \\lVert \\mathbf{n}(t) \\rVert_2<\\infty \\} \\right)=0,\\label{MAX-P1}\\\\\n\t\t&\t\\limsup_{t\\toup\\tilde{\\tau}_\\infty } \\lvert \\nabla \\v(t)\\rvert_{\\mathrm{L}^2}^2 +\\lVert \\d(t)\\rVert^2_2+\\int_0^t \\left(\\lvert \\mathrm{A}\\v(s)\\rvert^2_{\\mathrm{L}^2}+ \\lVert \\d(s) \\rVert^2_3 \\right)ds=\\infty \\text{ $\\mathbb{P}$-a.s. on } \\{\\tilde{\\tau}_\\infty<\\infty \\}\\label{MAX-P2}.\n\t\t\t\\end{align}\n\t\t\\end{enumerate}\n\t\t\n\t\\end{thm}\n\t\\begin{proof}\n\t\tLemma \\ref{SEM-1}-\\ref{SEM-3} show that $\\{\\mathbb{S}(t)\\}_{t\\geq 0}$ on\n\t\t$\\mathscr{H}=\\mathrm{H}\\times \\mathbf{X}_{0}$ satisfies Assumption \\ref{assum-01}.\n\t\tThanks to Proposition \\ref{SLC-ST} we can infer by applying Theorems \\ref{Thm:LocalUniqueness},\n\t\t\\ref{thm_local} and \\ref{thm_maximal-abstract} that problem\n\t\t\\eqref{ABSTRACT-LC} has a unique maximal local solution satisfying items \\eqref{item-1} and \\eqref{THM-ii} of Theorem \\ref{LC-Local-Sol}.\n\t\\end{proof}\n\t\\subsection{Existence and uniquness of global strong solution: 2D case}\nBy using the Khashminskii test for non-explosions, see \\cite[Theorem\nIII.4.1]{Kh_1980}, and some arguments from \\cite{Brz+Masl+Seidler_2005}, we prove in this section that if $d=2$ then the problem \\eqref{ABSTRACT-LC} has a unique global solution. \n\t\n\tFor all $(\\mathbf{u}, \\mathbf{d}) \\in C([0,T]; \\mathrm{H}\\times \\mathbf{H}^1) \\bigcap L^2(0,T; \\mathrm{V}\\times \\mathbf{H}^2)$ and $t\\in [0,T]$ we put \n\t\\begin{align}\n\t& \\mathcal{E}[\\mathbf{u},\\mathbf{d}](t)= \\frac{1}{2}\\left(\\lvert \\mathbf{u} (t) \\rvert^2_{\\mathrm{L}^2}+ \\lvert \\mathbf{d}(t)\\rvert_{\\mathbf{L}^2} + \\lvert \\nabla \\mathbf{d}(t) \\rvert^2_{\\mathbf{L}^2} + \\int_\\mathcal{O} F( \\mathbf{d}(t,x)) dx \\right)\\\\\n\t& \\mathscr{D}[\\mathbf{u}, \\mathbf{d} ](t) = \\lvert \\mathrm{A}^\\frac12 \\mathbf{u} (t) \\rvert^2_{\\mathrm{L}^2}+ \\lvert \\mathrm{A}_1\\mathbf{d}(t) - f(\\mathbf{d}(t) ) \\rvert^2_{\\mathbf{L}^2}.\n\t\\end{align}\n\n\t\n\t\\begin{thm}\\label{GLOBAL-ST}\n\t\tLet $d=2$, $N\\in \\mathbb{N}$, $\\mathbf{h} \\in \\mathbf{H}^2$ and $(\\v_0, \\d_0) \\in \\mathrm{L}^2(\\Omega; \\mathrm{V} \\times \\mathbf{X}_{1})$ such that \n\t\t\\begin{equation}\\label{E_0}\n\t\t\\mathbb{E} \\lvert \\mathcal{E}[\\v_0,d_0]\\rvert^{2(4N+2)})= \\mathbb{E} \\biggl(\\vert \\mathbf{v}_0\\vert^2_{\\mathrm{L}^2} + \\vert \\mathbf{n}_0\\vert^2_{\\mathbf{L}^2}+\\vert \\nabla \\mathbf{n}_0 \\vert^2_{\\mathbf{L}^2} + \\int_{\\mathcal{O}}F(\\mathbf{n}_0(x)) dx \\biggr)^{2(4N+2)}<\\infty.\n\t\t\\end{equation}\n\tIf Assumptions \\ref{eqn-f}, \\ref{Assum:Usual hypotheses} and \\ref{HYPO-ST}, \n\t\tthen the problem \\eqref{ABSTRACT-LC} has a unique global strong solution.\n\t\\end{thm}\n\tThe proof of this theorem is given at the end of this subsection.\n\n\t\\begin{prop}\\label{EST1}\n\t\tLet all the assumptions of Theorem \\ref{GLOBAL-ST} be satisfied and $p \\in [2, 2(4N+1)]$. Also, let $(\\tau_k)_{k \\in \\mathbb{N}}$ be the sequence of stopping times defined by\n\t\t\t\\begin{equation}\\label{STOP}\n\t\t\t\t\t\\tau_k=\\inf\\{t \\in [0,\\infty) : \\lvert \\nabla \\v(t)\\rvert_{\\mathrm{L}^2}^2 +\\lVert \\d(t)\\rVert^2_2+\\int_0^t \\left(\\lvert \\mathrm{A}\\v(s)\\rvert^2_{\\mathrm{L}^2}+ \\lVert \\d(s) \\rVert^2_3 \\right)ds >\n\t\t\t\t\tk^2\\}, \\; k\\in \\mathbb{N}.\n\t\t\t\t\t\\end{equation}\n\t\tThen, there exist an increasing function $\\varphi: [0,\\infty) \\to (0,\\infty)$ and $\\kappa_0=\\kappa_0(p,\\lvert \\mathbf{h} \\rvert_{\\mathbf{W}^{1,4}})>0$ such that for all $k \\in \\mathbb{N}$\n\t\t\\begin{equation}\\label{Eq:ESTofVDinWeakerNorm}\n\t\t\\mathbb{E} \\sup_{t\\in [0,T]}\\left( \\mathcal{E}[\\v,\\d] (t\\wedge \\tau_k) \\right)^p + \\mathbb{E} \\left[\\int_0^{T\\wedge \\tau_k} \\left(\\mathscr{D}[\\v,\\d](s) -\\frac{a_{N+1}}{2} \\lvert \\d(s) \\rvert^{2N+2}_{\\mathbf{L}^{2N+2}} \\right)\\, ds \\right] \\le \\kappa_0 \\varphi(T) \\Big( 1 + \\mathbb{E}\\lvert \\mathcal{E}[\\v,\\d](0)\\rvert^p \\Big).\n\t\t\\end{equation}\n\t\\end{prop}\n\t\\begin{proof}\n\t\tThe proof of this proposition will be given in Section \\ref{AppB}.\n\t\\end{proof}\n\tHereafter, we set\n\t\\begin{equation}\\label{Eq:ConstantFrakC0}\n\t\\mathfrak{C}_0= \\kappa_0 \\varphi(T)(1 + \\mathbb{E} \\lvert \\mathcal{E}(\\v,\\d)(0)\\rvert^{2(4N+2)} )\n\t\\end{equation}\n\t\\begin{cor}\n\t\tLet all the assumptions of Proposition \\ref{EST1} be satisfied. Then, there exists $C>0$ such that for all $k \\in \\mathbb{N}$\n\t\t\\begin{equation}\\label{Eq:EstIntegralOfH2Norm}\n\t\t\\mathbb{E} \\left[\\int_0^{T\\wedge \\tau_k}\\lVert \\d(s) \\rVert^2_2 \\right]^2 \\le C (\\mathfrak{C}_0 +1).\n\t\t\\end{equation}\n\t\\end{cor}\n\t\\begin{proof}\n\t\tBy \\eqref{bigdanh2} and $\\mathbf{H}^1 \\hookrightarrow \\mathbf{L}^{4N+2}$, which is valid for $d=2$, there exists a constant $C>0$ such that\n\t\t\\begin{equation}\\label{Eq:Bigdanh2withH1norm}\n\t\t\\lVert \\d \\rVert^2_2 \\le C( \\lvert \\mathrm{A}_1 \\d -f(\\d)\\rvert^2_{\\mathbf{L}^2}+ \\lVert \\d \\rVert^{4N+2}_1+1 ),\n\t\t\\end{equation}\n\t\tfrom which along with \\eqref{Eq:ESTofVDinWeakerNorm} we conclude the proof of the corollary.\n\t\\end{proof}\n\tLet $\\Psi_1: \\mathbf{H}^2 \\to [0, \\infty)$, $\\Psi_2: D(\\mathrm{A}) \\to [0,\\infty)$ and $\\Psi: D(\\mathrm{A})\\times \\mathbf{H}^2 \\to 0,\\infty) $ be defined by\n\t\\begin{align}\n\t& \\Psi_1(\\mathbf{d})= \\frac12 \\lvert \\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d}) \\rvert^2_{\\mathbf{L}^2}, \\;\\; \\mathbf{d}\\in \\mathbf{H}^2,\\label{Eq:DefPsi1forD} \\\\\n\t& \\Psi_2(\\mathbf{u})= \\frac12 \\lvert \\nabla \\mathbf{u} \\rvert^2_{\\mathrm{L}^2},\\;\\; \\mathbf{u}\\in D(\\mathrm{A}),\\label{Eq:DefPsi2forv}\\\\\n\t& \\Psi(\\mathbf{u},\\mathbf{d})=\\Psi_1(\\mathbf{d}) + \\Psi_2(\\mathbf{u}), \\text{ } (\\mathbf{u},\\mathbf{d})\\in D(\\mathrm{A}) \\times \\mathbf{H}^2.\\label{Eq:DefFunctionForIto}\n\t\\end{align}\n\tHereafter, $\\Psi_i^\\prime$ and $\\Psi_i^{\\prime\\prime}$, $i=1,2$, are the first and second Fr\\'echet derivatives of $\\Psi_i$, $i=1,2$.\n\n\t\\begin{lem}\\label{Lem:FirstDerAppliedtoVdotNabalaD}\n\t\tThere exists $\\kappa_1>0$ such that for all $\\mathbf{d}\\in \\mathbf{H}^3$ and $\\mathbf{u}\\in D(\\mathrm{A})$ we have\n\t\t\\begin{equation}\\label{Eq:FirstDerAppliedtoVdotNabalaD}\n\t\t-\t\\Psi_1^\\prime(\\mathbf{d})[\\mathbf{u}\\cdot \\nabla \\mathbf{d}] \\le \\kappa_1\\Psi(\\mathbf{u},\\mathbf{d}) \\left[\\lVert \\mathbf{d} \\rVert^2_{1} + 1\\right] \\lVert \\mathbf{d} \\rVert^2_{2} + \\frac14 \\lvert \\mathrm{A} \\mathbf{u} \\rvert^2_{\\mathrm{L}^2} + \\frac16 \\lvert \\nabla (\\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d})) \\rvert^2_{\\mathbf{L}^2}.\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tIn this proof $C>0$ is an universal constant which may change from one term to the other.\n\t\tLet $\\mathbf{d}\\in \\mathbf{H}^3\\cap D(\\mathrm{A}_1)$ and $\\mathbf{u}\\in D(\\mathrm{A})$. \n\t\tObserve that \n\t\t\\begin{align}\n\t\t\\Psi_1^\\prime(\\mathbf{d})[\\mathbf{g}]=\\langle \\Delta \\mathbf{d} -f(\\mathbf{d}), \\mathrm{A}_1 \\mathbf{g} + f^\\prime(\\mathbf{d})[\\mathbf{g}] \\rangle \\text{ for all } \\mathbf{g}\\in \\mathbf{H}^2, \\label{Eq:FirstDeriveOfPsi}\n\t\t\\end{align}\n\t\tand \n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t-\\Delta(\\mathbf{u}\\cdot \\nabla \\mathbf{d})=-(\\Delta \\mathbf{u} \\cdot \\nabla) \\mathbf{d} - (\\mathbf{u}\\cdot \\nabla) \\Delta \\mathbf{d} -2 \\mathrm{tr}(\\nabla \\mathbf{u} \\nabla^2) \\mathbf{d}\\\\\n\t\t=-(\\Delta \\mathbf{u} \\cdot \\nabla) \\mathbf{d} - 2 \\mathrm{tr}(\\nabla \\mathbf{u} \\nabla^2) \\mathbf{d}+ (\\mathbf{u}\\cdot \\nabla)[\\Delta \\mathbf{d} -f(\\mathbf{d})] - f^\\prime(\\mathbf{d})[\\mathbf{u}\\cdot \\nabla\\mathbf{d}]\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tHence, by using the identities \\eqref{tild-b-0} and \\eqref{Eq:FirstDeriveOfPsi} we obtain\n\t\t\\begin{equation}\n\t\t\\begin{split}\n\t\t-\t\\Psi_1^\\prime(\\mathbf{d})[ \\mathbf{u}\\cdot \\nabla \\mathbf{d}]=\\langle \\mathrm{A}_1\\mathbf{d}-f(\\mathbf{d}), (\\Delta \\mathbf{u}\\cdot \\nabla) \\mathbf{d} \\rangle +2\\langle \\mathrm{A}_1\\mathbf{d}-f(\\mathbf{d}), \\mathrm{tr}(\\nabla \\mathbf{u} \\nabla^2)\\mathbf{d}\\rangle,\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\twhich along with \\eqref{GAG-l4} and the H\\\"older and Young inequalities imply that\n\t\t\\begin{equation}\n\t\t\\begin{split}\n\t\t-\t\\Psi_1^\\prime(\\mathbf{d})[ \\mathbf{u}\\cdot \\nabla \\mathbf{d}]& \\le C \\lvert \\mathrm{A}_1 \\mathbf{d}-f(\\mathbf{d}) \\rvert_{\\mathbf{L}^4} [\\lvert \\mathrm{A} \\mathbf{u} \\rvert_{\\mathrm{L}^2} \\lvert \\nabla \\mathbf{d} \\rvert_{\\mathbf{L}^4} + \\lvert \\nabla \\mathbf{u} \\rvert_{\\mathrm{L}^4} \\lvert \\nabla^2 \\mathbf{d} \\rvert_{\\mathbf{L}^2} ]\\\\\n\t\t& \\le \\frac16 \\lvert \\nabla (\\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d}) ) \\rvert^2_{\\mathbf{L}^2}+ \\frac14 \\lvert \\mathrm{A} \\mathbf{u} \\rvert^2_{\\mathrm{L}^2}+ C \\lvert \\mathrm{A}_1 \\mathbf{d}-f(\\mathbf{d})\\rvert_{\\mathbf{L}^2}^2 [ \\lvert \\nabla \\mathbf{d} \\rvert_{\\mathbf{L}^4}^4 + \\lvert \\nabla^2 \\mathbf{d} \\rvert_{\\mathbf{L}^2} ].\n\t\t\\end{split}\n\t\t\\end{equation}\t\n\t\tNow, \\eqref{Eq:FirstDerAppliedtoVdotNabalaD} easily follows from the last line of the the above chain of inequalities.\n\t\\end{proof}\n\t\\begin{lem}\\label{Lem:FirstDerAppliedtoDeltaD+f(D)}\n\t\tThere exists $\\kappa_2>0$ such that for all $\\mathbf{d}\\in \\mathbf{H}^3\\cap D(\\mathrm{A}_1)$ and $\\mathbf{u}\\in D(\\mathrm{A})$ we have\n\t\t\\begin{equation}\\label{Eq:FirstDerAppliedtoDeltaD+f(D)}\n\t\\langle\tf ^\\prime(\\mathbf{d})[\\mathrm{A}_1\\mathbf{d}-f(\\mathbf{d})], \\mathrm{A}_1\\mathbf{d}-f(\\mathbf{d}) \\rangle\\le \\frac16 \\lvert \\nabla(\\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d})) \\rvert^2_{\\mathbf{L}^2} + \\kappa_2 \\Psi(\\mathbf{u},\\mathbf{d}) (1+ \\lVert \\mathbf{d} \\rVert^{4N}_1 ).\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tUsing part \\eqref{Itemi:REM-H2} of Remark \\ref{REM-H2}, \\eqref{GAG-l4}, the H\\\"older and Young inequalities, and by $\\mathbf{H}^1\\hookrightarrow \\mathrm{L}^{4N}$ (that is valid for $d=2$) we infer that there exist $C>0$ and $\\kappa_2>0$ such that\n\t\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\t\\langle\tf ^\\prime(\\mathbf{d})[\\mathrm{A}_1\\mathbf{d}-f(\\mathbf{d})], \\mathrm{A}_1\\mathbf{d} -f(\\mathbf{d}) \\rangle\n\t\t\t&\\le c_1 \\int_\\mathcal{O} (1+ \\lvert \\mathbf{d} \\rvert^{2N}) \\lvert \\mathrm{A}_1\\mathbf{d} -f(\\mathbf{d})\\rvert^2 dx\\\\\n\t\t\t&\\le C \\lvert \\mathrm{A}_1\\mathbf{d}-f(\\mathbf{d}) \\rvert_{\\mathrm{L}^4}^2 \\lvert (1+ \\lvert \\mathbf{d} \\rvert^{2N}) \\rvert^2_{\\mathbf{L}^2}\\\\\n\t\t&\\le \\frac16 \\lvert \\nabla (\\mathrm{A}_1\\mathbf{d}-f(\\mathbf{d})) \\rvert^2_{\\mathbf{L}^2}+ \\kappa_2 \\lvert \\mathrm{A}_1\\mathbf{d}-f(\\mathbf{d}) \\rvert_{\\mathrm{L}^2}^2 (1 + \\lVert\\mathbf{d}\\rVert^{4N}_{1}),\n\t\t\\end{split}\n\t\t\\end{equation}\t\n\t\tfor all $\\mathbf{d}\\in \\mathbf{H}^3\\cap D(\\mathrm{A}_1)$ and $\\mathbf{u}\\in D(\\mathrm{A})$. This completes the proof of the lemma.\n\t\\end{proof}\n\t\\begin{lem}\\label{Lem:SecondDerofPsiAppliedtoGG}\n\t\tLet $\\mathbf{h}\\in \\mathbf{H}^2$. Then, there exists $\\kappa_5=\\kappa_5(\\lVert \\mathbf{h} \\rVert_2)>0$ such that \n\t\t\\begin{equation}\n\t\t\\lvert\t\\Psi_1^{\\prime \\prime}(\\mathbf{d})[\\mathbf{d}\\times \\mathbf{h}, \\mathbf{d}\\times \\mathbf{h}] \\rvert+\\lvert\t\\Psi_1^\\prime(\\mathbf{d})[(\\mathbf{d}\\times\\mathbf{h})\\times \\mathbf{h}] \\rvert \\le \\Psi_1(\\mathbf{d})+ \\kappa_5 (1+ \\lVert \\mathbf{d} \\rVert^{4N}_{1}) \\lVert \\mathbf{d} \\rVert^2_{2},\\text{ for all $ \\mathbf{d} D(\\mathrm{A}_1)$.}\n\t\t\\label{Eq:SecondDerofPsiAppliedtoGG}\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tLet \\text{ $(\\mathbf{u}, \\mathbf{d})\\in D(\\mathrm{A}) \\times D(\\mathrm{A}_1)$.}\n\t\tWe firstly recall that \n\t\t\\begin{equation}\n\t\t\\Psi_1^{\\prime \\prime}(\\mathbf{d})[\\mathbf{g}, \\mathbf{p}]=\\langle \\mathrm{A}_1 \\mathbf{d}-f(\\mathbf{d}), -f^{\\prime \\prime}(\\mathbf{d})[\\mathbf{g}, \\mathbf{p}] \\rangle + \\langle \\mathrm{A}_1 \\mathbf{p} - f^{\\prime}(\\mathbf{d})[\\mathbf{p}], \\mathrm{A}_1 \\mathbf{g}- f^\\prime(\\mathbf{d})[\\mathbf{g}]\\rangle,\\; \\mathbf{p}, \\mathbf{g} \\in D(\\mathrm{A}_1).\n\t\t\\end{equation}\n\t\tHence, by recalling that $G(\\mathbf{d})=\\mathbf{d}\\times \\mathbf{h}$ we have \n\t\t\\begin{align}\n\t\t\\Psi_1^{\\prime \\prime}(\\mathbf{d})[G(\\mathbf{d}),G(\\mathbf{d})]&= \\lvert \\mathrm{A}_1(G(\\mathbf{d}))- f^\\prime(\\mathbf{d})[G(\\mathbf{d})]\\rvert^2_{\\mathbf{L}^2}+ \\langle \\mathrm{A}_1\\mathbf{d} -f(\\mathbf{d}), \\mathrm{A}_1(G(\\mathbf{d}))- f^{\\prime\\prime}(\\mathbf{d})[G(\\mathbf{d}), G(\\mathbf{d})]\\rangle\n\t\\nonumber \t\\\\\n\t\t&\\le \\frac14 \\left\\lvert\\mathrm{A}_1 \\mathbf{d}-f(\\mathbf{d})\\right\\rvert^2_{\\mathbf{L}^2}+\\left\\lvert \\mathrm{A}_1(G(\\mathbf{d}))-f^{\\prime\\prime}(\\mathbf{d})[G(\\mathbf{d}), G(\\mathbf{d})]\\right\\rvert^2_{\\mathbf{L}^2} + \\lvert \\mathrm{A}_1(G(\\mathbf{d}))-f^\\prime(\\mathbf{d})[G(\\mathbf{d})]\\rvert^2_{\\mathbf{L}^2}\\nonumber \\\\\n\t\t& =\\mathrm{I}_1+ \\mathrm{I}_2+\\mathrm{I}_3\\label{Eq:EstDerA}.\n\t\t\\end{align}\n\t\tSecondly, by the part \\eqref{Itemi:REM-H2} of Remark \\ref{REM-H2}, $\\mathbf{H}^2\\hookrightarrow\\mathbf{L}^\\infty$, $\\mathbf{H}^1 \\hookrightarrow \\mathrm{L}^4, \\mathbf{L}^{4N}$ and the H\\\"older and Young inequalities we infer that there exists a constant $C=C(\\lVert \\mathbf{h} \\rVert_2 )>0$ such that \n\t\t\\begin{equation}\\label{Eq:EstDerB}\n\t\t\\begin{split}\n\t\t\\mathrm{I}_2 \\le & \\lvert \\Delta(\\mathbf{d}\\times \\mathbf{h}) \\rvert^2_{\\mathbf{L}^2} + \\lvert \\lvert \\mathbf{d} \\times \\mathbf{h} \\rvert^2 \\lvert f^{\\prime\\prime}(\\mathbf{d})\\rvert \\rvert^2_{\\mathbf{L}^2}\\\\\n\t\t&\\le 4 \\left(\\lvert \\Delta \\mathbf{d}\\times \\mathbf{h} \\rvert^2_{\\mathbf{L}^2}+ 2 \\lvert \\nabla \\mathbf{d} \\times \\nabla \\mathbf{h} \\rvert^2_{\\mathbf{L}^2} + \\lvert \\mathbf{d} \\times \\Delta \\mathbf{h} \\rvert^2_{\\mathbf{L}^2}\\right) + C (1+ \\lvert \\mathbf{d} \\rvert^{4N}_{\\mathbf{L}^{4N}} )\\lvert \\mathbf{d}\\times \\mathbf{h}\\rvert_{\\mathbf{L}^\\infty}^2 \\\\\n\t\t&\\le C \\lVert \\mathbf{d} \\rVert^2_2(1+ \\lVert \\mathbf{d}\\rVert^{4N}_1 ).\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tIn a similar way, we prove that there exists a constant $C_7=C(\\lVert \\mathbf{h} \\rVert_2 )>0$ such that \n\t\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\\mathrm{I}_3 \\le C \\lVert \\mathbf{d} \\rVert^2_2(1+ \\lVert \\mathbf{d}\\rVert^{4N}_1 ).\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tCombining this last inequality with \\eqref{Eq:EstDerA} and \\eqref{Eq:EstDerB} proves that $$\\lvert \\Psi_1^{\\prime \\prime}(\\mathbf{d})[G(\\mathbf{d}),G(\\mathbf{d})]\\rvert \\le \\frac12 \\Psi_1(\\mathbf{d}) + C_7 \\lVert \\mathbf{d} \\rVert^2_2(1+ \\lVert \\mathbf{d}\\rVert^{4N}_1 ).$$ In a similar way, we can also show that \n\t\t$\\lvert \\Psi_1^{\\prime}(\\mathbf{d})[(\\mathbf{d}\\times \\mathbf{h})\\times \\mathbf{h}]\\rvert \\le \\frac12 \\Psi_1(\\mathbf{d}) + C_7 \\lVert \\mathbf{d} \\rVert^2_2(1+ \\lVert \\mathbf{d}\\rVert^{4N}_1 ).$ One easily conclude the proof of the lemma from the last two estimates.\n\t\\end{proof}\n\t\\begin{lem}\\label{Lem:Psi1BDGNeeded}\n\t\tLet $\\mathbf{h}\\in \\mathbf{H}^2$. Then, there exists $\\kappa_5=\\kappa_5(\\lVert \\mathbf{h} \\rVert_2)>0$ such that for all $\\mathbf{d} \\in D(\\mathrm{A}_1)$ \n\t\t\\begin{equation}\\label{Eq:Psi1BDGNeeded}\n\t\t\\lvert \\Psi_1^{\\prime}(\\mathbf{d})[\\mathbf{d}\\times \\mathbf{h}] \\rvert\\le \\kappa_6 \\left[1+ \\Psi_1(\\mathbf{d}) + \\lVert \\mathbf{d} \\rVert^{4N+1}_{1} +\\lVert \\mathbf{d} \\rVert_1 \\lVert \\mathbf{d}\\rVert_2 \\right].\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tBy part \\eqref{Itemi:REM-H2} of Remark \\ref{REM-H2}, \\eqref{GAG-l4}, \\eqref{GAG-LInf}, \\eqref{GAG-LInf-2}, the H\\\"older and Young inequalities and $\\mathbf{H}^1\\hookrightarrow \\mathbf{L}^{4N+2}, \\mathbf{L}^4$, we infer that there exists $\\kappa_6=\\kappa_6(\\lVert \\mathbf{h} \\rVert_2)>0$ such that for all $\\mathbf{d} \\in D(\\mathrm{A}_1)$ \n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\lvert \\Psi_1^{\\prime}(\\mathbf{d})[\\mathbf{d}\\times \\mathbf{h}] \\rvert&\\le\n\t\t\\lvert \\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d})\\rvert_{\\mathbf{L}^2} \\left(\\lvert [\\mathrm{A}_1\\mathbf{d} -f(\\mathbf{d})]\\times \\mathbf{h} + 2 \\nabla \\mathbf{d} \\times \\nabla \\mathbf{h} + \\mathbf{d} \\times \\Delta \\mathbf{h} -f(\\mathbf{d}) \\times \\mathbf{h} \\rvert_{\\mathbf{L}^2}\\right)\\\\\n\t\t&\t\\le \\kappa_6 \\left( \\Psi_1(\\mathbf{d})+ \\lvert \\nabla \\mathbf{d} \\rvert^2_{\\mathbf{L}^4} + \\lvert \\mathbf{d} \\rvert^2_{\\mathbf{L}^\\infty} + \\lvert \\mathbf{d} \\rvert^{4N+2}_{\\mathbf{L}^{4N+2}} +1\\right),\\\\\n\t\t&\t\\le \\kappa_6 \\left( \\Psi_1(\\mathbf{d})+ \\lVert \\mathbf{d} \\rVert_{1} \\lVert \\mathbf{d} \\rVert_{2} + \\lvert \\mathbf{d} \\rvert^{4N+2}_{1} +1 \\right).\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tThis completes the proof of the Lemma \\ref{Lem:Psi1BDGNeeded}.\n\t\\end{proof}\n\tWe will also need to the following results.\n\t\\begin{lem}\\label{Lem:NonlienarNSeItoV}\n\t\tThere exists $\\kappa_3>0$ such that for all $\\mathbf{u}\\in D(\\mathrm{A})$ and $\\mathbf{d}\\in D(\\mathrm{A}_1)$ \n\t\t\\begin{equation}\n\t\t-\\Psi^\\prime_2(\\mathbf{u})(B(\\mathbf{u},\\mathbf{u}))=\t-\\langle B(\\mathbf{u},\\mathbf{u}), \\mathrm{A} \\mathbf{u}\\rangle \\le \\frac14 \\lvert \\mathrm{A} \\mathbf{u} \\rvert^2_{\\mathrm{L}^2} + \\kappa_3 ( \\lvert \\mathbf{u}\\rvert^2_{\\mathrm{L}^2} \\lvert \\nabla \\mathbf{u} \\rvert^2_{\\mathrm{L}^2}) \\Psi(\\mathbf{u},\\mathbf{d}).\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tUsing \\eqref{GAG-l4}, the H\\\"older and Young inequalities we infer that $C>0$ such that\n\t\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\\langle B(\\mathbf{u},\\mathbf{u}) , \\mathrm{A}\\mathbf{u} \\rangle &=\\langle \\mathbf{u}\\cdot \\nabla \\mathbf{u}, \\mathrm{A}\\mathbf{u}\\rangle\n\t\t\\le \\lvert \\mathrm{A} \\mathbf{u} \\rvert \\lvert \\mathbf{u} \\rvert_{\\mathrm{L}^4} \\lvert \\nabla \\mathbf{u} \\rvert_{\\mathrm{L}^4}\\\\\n\t\t&\t\\le \\lvert \\mathrm{A} \\mathbf{u} \\rvert^\\frac32 \\lvert \\mathbf{u} \\rvert^\\frac12_{\\mathrm{L}^2} \\lvert \\nabla \\mathbf{u} \\rvert^\\frac12_{\\mathrm{L}^2}\n\t\t\t\\le \\frac14 \\lvert \\mathrm{A} \\mathbf{u} \\rvert^2_{\\mathrm{L}^2} + C \\lvert \\mathbf{u} \\rvert^2_{\\mathrm{L}^2} \\lvert \\nabla \\mathbf{u}\\rvert^2_{\\mathrm{L}^2} \\Psi_1(\\mathbf{u}),\n\t\t\\end{split}\n\t\t\\end{equation}\n\t for all $(\\mathbf{u}, \\mathbf{d})\\in D(\\mathrm{A})\\times D(\\mathrm{A}_1)$. We easily conclude the proof of Lemma \\ref{Lem:NonlienarNSeItoV} from last line.\n\t\\end{proof}\n\t\\begin{lem}\\label{Lem:CouplingTermIto}\n\t\tThere exists $\\kappa_4>0$ such that \n\t\t\\begin{equation}\n\t\t-\\Psi^\\prime_2(\\mathbf{u})(M(\\mathbf{d},\\mathbf{d}))=\t-\\langle M(\\mathbf{d},\\mathbf{d}), \\mathrm{A}\\mathbf{u}\\rangle \\le \\frac14 \\lvert \\mathrm{A} \\mathbf{u}\\rvert^2_{\\mathrm{L}^2} + \\frac16 \\lvert \\nabla (\\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d})) \\rvert^2_{\\mathbf{L}^2} + \\kappa_4 \\Psi(\\mathbf{u},\\mathbf{d}) \\lVert \\mathbf{d} \\rVert^2_1 \\lVert \\mathbf{d} \\rVert^2_2,\n\t\t\\end{equation}\n\t\tfor all $\\mathbf{d}\\in D(\\mathrm{A}_1)$ satisfying $\\nabla (\\mathrm{A}_1 \\mathbf{d} +f(\\mathbf{d})) \\in \\mathrm{L}^2$, and $\\mathbf{u}\\in D(\\mathrm{A})$.\n\t\\end{lem}\n\t\\begin{proof}\n\t\tIn this proof $C>0$ is an universal constant.\n\t\tLet $\\mathbf{d}\\in D(\\mathrm{A}_1)$ be such that $\\nabla (\\mathrm{A}_1 \\mathbf{d} +f(\\mathbf{d})) \\in \\mathrm{L}^2$, and $\\mathbf{u}\\in D(\\mathrm{A})$.\n\t\tFirstly, since $\\Pi: \\mathrm{L}^2 \\to \\mathrm{H}$ is self-adjoint, $\\nabla F(\\mathbf{d})=\\nabla \\mathbf{d} f(\\mathbf{d})$ and $\\Div \\mathrm{A} \\mathbf{u}=0$, we infer that\n\t\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\\langle M(\\mathbf{d},\\mathbf{d}) , \\mathrm{A}\\mathbf{u} \\rangle=& \\frac12\\langle \\mathrm{A}\\mathbf{u}, \\nabla \\lvert \\nabla \\mathbf{d}\\rvert^2\\rangle -\\langle \\mathrm{A} \\mathbf{u}, \\nabla \\mathbf{d} \\Delta \\mathbf{d}\\rangle \\\\\n\t\t=&- \\langle \\mathrm{A} \\mathbf{u} \\cdot\\nabla \\mathbf{d}, \\mathrm{A}_1 \\mathbf{d}+f(\\mathbf{d})\\rangle + \\langle \\mathrm{A} \\mathbf{u} , \\nabla F(\\mathbf{d})\\rangle.\n\t\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tSecondly, applying the H\\\"older, the Gagliardo-Nirenberg and the Young inequality yields\n\t\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\\langle M(\\mathbf{d},\\mathbf{d}) , \\mathrm{A}\\mathbf{u} \\rangle=& -\\langle \\mathrm{A} \\mathbf{u} \\cdot\\nabla \\mathbf{d}, \\mathrm{A}_1 \\mathbf{d}+f(\\mathbf{d})\\rangle\n\t\t\\le \\lvert \\mathrm{A} \\mathbf{u} \\rvert_{\\mathrm{L}^2} \\lvert \\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d}) \\rvert_{\\mathbf{L}^4} \\lvert \\nabla \\mathbf{d} \\rvert_{\\mathrm{L}^4}\\\\\n\t\t\\le &\\frac14 \\lvert \\mathrm{A} \\mathbf{u} \\rvert^2_{\\mathrm{L}^2}+ \\frac16 \\lvert \\nabla (\\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d})) \\rvert^2_{\\mathbf{L}^2} + C \\lvert \\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d}) \\rvert^2_{\\mathbf{L}^2} \\lVert \\mathbf{d} \\rVert^2_1 \\lVert \\mathbf{d} \\rVert^2_2.\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tThis completes the proof of Lemma \\ref{Lem:CouplingTermIto}\n\t\\end{proof}\nNow, let $\\kappa_i$, $i=1,\\ldots,4$ be the constants from Lemmata \\ref{Lem:FirstDerAppliedtoVdotNabalaD}, \\ref{Lem:FirstDerAppliedtoDeltaD+f(D)}, \\ref{Lem:NonlienarNSeItoV} and \\ref{Lem:CouplingTermIto}. For all $t\\ge 0 $ we set\n\t\\begin{equation}\\label{Eq:WeightToCancelbadterms}\n\t\\begin{split}\n\t\\Phi(t)=\\exp\\left({-\\int_0^{t}\\left[ (\\kappa_1 +\\kappa_4)(1+ \\lVert \\d(s) \\rVert^2_{1}) \\lVert \\d(s) \\rVert^2_2 + \\kappa_2 (1+ \\lVert \\d(s) \\rVert^{4N}_1) + \\kappa_3 \\lvert \\v(s) \\rvert^2_{\\mathrm{L}^2} \\lvert \\nabla \\v(s) \\rvert^2_{\\mathrm{L}^2} \\right] ds }\\right).\n\t\\end{split}\n\t\\end{equation}\n\n\n\n\n\tLet $\\mathfrak{C}_0>0$ be the constant defined in \\eqref{Eq:ConstantFrakC0} and $\\mathfrak{C}_1>0$ the constant defined by\n\t\\begin{equation}\\label{Eq:ConstantFrakC1}\n\t\\mathfrak{C}_1= \\Psi(\\v_0,\\d_0) + \\mathfrak{C}_0+1\n\t\\end{equation}\n\t\\begin{prop}\\label{STRONGER-NORM}\n\t\tLet $\\Psi_1$, $\\Phi$, $\\mathfrak{C}_0$ and $\\mathfrak{C}_1$ be defined in \\eqref{Eq:DefPsi1forD}, \\eqref{Eq:WeightToCancelbadterms}, \\eqref{Eq:ConstantFrakC0} and \\eqref{Eq:ConstantFrakC1}, respectively. Let \t$(\\tau_k)_{k \\in \\mathbb{N}}$ be the sequence of stopping times defined in \\eqref{STOP}. \tLet $d=2$, $N\\in \\mathbb{N}$ and $\\mathbf{h} \\in \\mathbf{H}^2$. \n\t\t\n\t\tIf all the other assumptions of Theorem \\ref{GLOBAL-ST} are satisfied, then there exists an increasing function $\\psi:[0,\\infty) \\to (0,\\infty)$ and $\\kappa_9=\\kappa_9(N, \\lVert \\mathbf{h} \\rVert_2)>0$ such that for all $k \\in \\mathbb{N}$\n\t\t\\begin{align}\n\t\t&\t\\mathbb{E} \\sup_{s\\in[0,T]} \\Phi(s\\wedge \\tau_k)\\left(\\lvert \\nabla \\v(s\\wedge \\tau_k) \\rvert^2_{\\mathrm{L}^2}+\\Psi_1(\\d(s\\wedge \\tau_k)) \\right)\\le \\kappa_9 \\psi(T) \\mathfrak{C}_1,\\label{Eq:EstSupofDel+F}\\\\\n\t\t& \\mathbb{E} \\int_0^{T\\wedge \\tau_k} \\Phi(s) \\left( \\lvert \\mathrm{A}\\v(s)\\rvert^2_{\\mathrm{L}^2} +\\lvert \\nabla (\\mathrm{A}_1\\d(s)+f(\\d(s) )\\rvert^2_{\\mathbf{L}^2} \\right)\\, ds \\le \\kappa_9 \\psi(T) \\mathfrak{C}_1. \\label{Eq:EstInteofNablaDelta+F}\n\t\t\\end{align}\n\t\\end{prop}\n\t\\begin{proof}\n\t\tThe proof of this proposition will be given in Section \\ref{AppB}.\n\t\\end{proof}\n\t\\begin{cor}\n\t\tUnder all the assumptions of Proposition \\ref{STRONGER-NORM}, there exists a $C>0$ such that for all $k \\in \\mathbb{N}$\n\t\t\\begin{equation}\\label{Eq:EstIntegrofH3NormofD}\n\t\t\\mathbb{E} \\int_0^{T\\wedge \\tau_k} \\Phi(s) \\lVert \\d(s) \\rVert^2_{3} ds\\le C(\\mathfrak{C}_0 + \\mathfrak{C}_1+1).\n\t\t\\end{equation}\n\t\\end{cor}\n\t\\begin{proof}\n\t\tBy part \\eqref{Itemi:REM-H2} of Remark \\ref{REM-H2}, the H\\\"older inequality and $\\mathbf{H}^1\\hookrightarrow \\mathbf{L}^{8N}\\hookrightarrow \\mathbf{L}^4$ we infer that\n\t\t\\begin{align*}\n\t\t\\lvert \\nabla f(\\d) \\rvert^2_{\\mathbf{L}^2} = & \\lvert f^\\prime(\\d)[\\nabla \\d] \\rvert^2_{\\mathbf{L}^2}\\le C(( \\lvert 1+ \\lvert \\d \\rvert^{2N}) \\lvert \\nabla \\d \\rvert \\rvert^2_{\\mathbf{L}^2} )\\\\\n\t\t\\le & \\lvert \\nabla \\d\\rvert^2_{\\mathbf{L}^4}+ \\lvert \\d \\rvert^{4N}_{\\mathbf{L}^{8N}} \\lvert \\nabla \\d \\rvert^2_{\\mathbf{L}^4}\n\t\t\\le C (\\lVert \\d \\rVert^2_{2} + \\lVert \\d \\rVert_1^{8N+2}).\n\t\t\\end{align*}\n\t\tWith this at hand we complete the proof by using \\eqref{Eq:EstIntegralOfH2Norm}, \\eqref{Eq:EstInteofNablaDelta+F} and the fact $$\\lVert \\d \\rVert^2_3\\le \\lVert \\d\\rVert^2_2 + 2 \\lvert \\nabla (\\Delta \\d +f(\\d)) \\rvert^2_{\\mathbf{L}^2} +2 \\lvert \\nabla f(\\d) \\rvert^2_{\\mathbf{L}^2}.$$\n\t\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\tAfter all these preparations we now proceed to the promised proof of Theorem \\ref{GLOBAL-ST}.\n\t\\begin{proof}[Proof of Theorem \\ref{GLOBAL-ST}]\n\t\tBy Theorem \\ref{LC-Local-Sol} the problem \\eqref{ABSTRACT-LC} has a unique maximal local solution $((\\mathbf{v},\\mathbf{n}); \\tilde{\\tau}_\\infty)$. We shall prove that $\\mathbb{P}\\Big(\\tilde{\\tau}_\\infty<\\infty \\Big)=0 $. For this aim, let $\\{ \\tau_k; k \\in \\mathbb{N}\\}$ be the sequence of stopping times defined in \\eqref{STOP}. \n\t\n\tWe first establish the following chain of inequalities\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\mathbb{P}\\left(\\tau_k k^2\\} } \\right),\\\\\n\t\t&\\le \\frac{1}{k^2}\\mathbb{E}\\biggl(1_{\\{\\tau_k 2 \\log{k}\\} } \\biggr)=:\\mathrm{I}+\\mathrm{II}.\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tNow, we estimate $\\mathrm{I}$ and $\\mathrm{II}$ separately.\n\t\tFirstly, from the definition of $\\tau_k$ and $\\Phi$ we have\n\t\t\\begin{align*}\n\t\t\\mathrm{I}\\le &\\frac{1}{k^2}\\mathbb{E} \\left[1_{\\{\\tau_k < t\\}} \\Phi(t\\wedge \\tau_k)\\left( \\lvert \\nabla \\mathbf{v}(t\\wedge \\tau_k) \\rvert^2_{\\mathrm{L}^2} + \\lVert \\mathbf{n}(t\\wedge \\tau_k) \\rVert^2_2 \\right) \\right]\\\\\n\t\t&\\qquad + \\frac1{k^2}\\mathbb{E} \\left[1_{\\{\\tau_k < t\\}} \\Phi(t\\wedge \\tau_k)\\int_0^{t\\wedge \\tau_k}\\left( \\lvert \\mathrm{A} \\mathbf{v}(s) \\rvert^2_{\\mathrm{L}^2} + \\lVert \\mathbf{n}(s) \\rVert^2_3 \\right)\\, ds \\right]\\\\\n\t\t\\le & \\frac{1}{k^2}\\mathbb{E} \\left[\\Phi(t\\wedge \\tau_k)\\left( \\lvert \\nabla \\mathbf{v}(t\\wedge \\tau_k) \\rvert^2_{\\mathrm{L}^2} + \\lVert \\mathbf{n}(t\\wedge \\tau_k) \\rVert^2_2\\right) \\right]+\\frac{1}{k^2}\\mathbb{E} \\left[\\int_0^{t\\wedge \\tau_k}\\Phi(s)\\left( \\lvert \\mathrm{A} \\mathbf{v}(s) \\rvert^2_{\\mathrm{L}^2} + \\lVert \\mathbf{n}(s) \\rVert^2_3 \\right)\\, ds \\right]\n\t\t\\end{align*}\n\t\tFrom \\eqref{Eq:Bigdanh2withH1norm}, \\eqref{Eq:ESTofVDinWeakerNorm} , \\eqref{Eq:EstInteofNablaDelta+F} and \\eqref{Eq:EstIntegrofH3NormofD} we infer that there exists a constant $C>0$ such that for all $k \\in \\mathbb{N}$\n\t\t\\begin{equation*}\n\t\t\\mathrm{I}\\le \\frac{1}{k^2}C (\\mathfrak{C}_0+\\mathfrak{C}_1+1).\n\t\t\\end{equation*}\n\t\tSecondly, we estimate $\\mathrm{II}$ as follows\n\t\t\\begin{align*}\n\t\t\\mathrm{II}=& \\mathbb{E}\\left(1_{\\{\\tau_k < t\\}} 1_{\\{\\int_0^{t\\wedge \\tau_k} \\phi(r)\\, dr > 2\\log{k} \\} }\\right)\n\t\t\\le \\int_{\\{\\tau_k2\\log{k} \\}}\\frac{ \\int_0^{t\\wedge \\tau_k} \\phi(r)\\, dr}{2\\log{k}} d\\mathbb{P}\\\\\n\t\t\\le & \\frac1{2\\log{k}} \\int_{\\{\\tau_k < t\\}} \\int_0^{t\\wedge \\tau_k} \\phi(r)\\, dr\\;\\;\\; d\\mathbb{P}\n\t\t\\le \\frac{1}{2\\log{k}} \\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\phi(r)\\, dr.\n\t\t\\end{align*}\n\t\tNow, from \\eqref{Eq:ESTofVDinWeakerNorm} and \\eqref{Eq:EstIntegralOfH2Norm} we infer that there exists a constant $C>0$ such that for all $k \\in \\mathbb{N}$\n\t\t\\begin{align*}\n\t\t&\\mathbb{E}\\int_0^{t\\wedge \\tau_k} \\lvert \\v(s)\\rvert^2_{\\mathrm{L}^2}\\lvert \\nabla \\v(s)\\rvert_{\\mathrm{L}^2}^2 ds\n\t\t\\le \\frac12 \\mathbb{E}\\sup_{0\\le s\\le t\\wedge \\tau_k}\\lvert \\v(s)\\rvert^4_{\\mathrm{L}^2} + \\frac12 \\mathbb{E} \\biggl[\\int_0^{t\\wedge \\tau_k}\n\t\t\\lvert \\nabla \\v(s)\\rvert_{\\mathrm{L}^2}^2 ds\n\t\t\\biggr]^2\\le C (\\mathfrak{C}_0+1),\\\\\n\t\t& \\mathbb{E}\\int_0^{t\\wedge \\tau_k}(1+1 \\lVert \\d(s)\\rVert^{2N}_{\\mathrm{H}^1})^2 ds\\le C (\\mathfrak{C}_0+1),\\\\\n\t\t& \\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\Vert \\mathbf{n}(s) \\Vert^2_1 \\Vert \\mathbf{n} (s) \\Vert^2_2 ds \\le \\frac12 \\mathbb{E} \\sup_{0\\le s\\le t\\wedge \\tau_k} \\Vert \\mathbf{n}(s)\\Vert^4_1 + \\frac12 \\mathbb{E} \\biggl[\\int_0^{t\\wedge \\tau_k} \\Vert \\mathbf{n}(s) \\Vert^2_2 ds \\biggr]^2\\le C (\\mathfrak{C}_0+1).\n\t\t\\end{align*}\n\t\tThus, there exists a constant $C>0$ such that for all $k\\in \\mathbb{N}$\n\t\t\\begin{equation*}\n\t\t\\mathrm{II} \\le \\frac1{2\\log{k}} \\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\phi(s)ds\\le \\frac{C (\\mathfrak{C}_0+1)}{\\log{k}}.\n\t\t\\end{equation*}\n\t\tCollecting the information about $\\mathrm{I}$ and $\\mathrm{II}$ together, we infer that\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\lim_{k\\rightarrow \\infty} \\mathbb{P}\\left(\\tau_k0$ such that for all $\\mathbf{d}\\in \\mathrm{ H}^1$ \n\t\t\\begin{equation}\\label{Eq:EstimateItoStratoCorrect}\n\t\t\\lvert \\nabla G(\\mathbf{d}) \\rvert^2_{\\mathbf{L}^2} + \\langle \\nabla \\mathbf{d}, \\nabla G^2(\\mathbf{d}) \\rvert \\le C \\lVert \\mathbf{d} \\rVert^2_{1}.\n\t\t\\end{equation}\n\t\\end{lem}\n\t\\begin{proof}\n\tLet $\\mathbf{h} \\in \\mathbf{W}^{1,4}$. Using the H\\\"older inequality, the embeddings $\\mathbf{H}^1 \\hookrightarrow \\mathbf{L}^4$ and $\\mathbf{W}^{1,4}\\hookrightarrow \\mathrm{L}^\\infty$, and\n\t\t\\begin{equation}\\label{Eq:MixedScalCrossProducts}\n\t\t\\mathbf{a} \\cdot [(\\mathbf{b} \\times \\mathbf{c}) \\times \\mathbf{d} ]= - (\\mathbf{a} \\times \\mathbf{d}) \\cdot (\\mathbf{b} \\times \\mathbf{c}), \\;\\; \\forall\\;\\; \\mathbf{a}, \\mathbf{b}, \\mathbf{c}, \\mathbf{d}\\in \\mathbb{R}^3,\n\t\t\\end{equation}\n\t\twe infer that for all $\\mathbf{d}\\in \\mathrm{ H}^1$ and $i \\in \\{1,2\\}$\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\lvert \\partial_i (\\mathbf{d}\\times \\mathbf{h}) \\rvert^2_{\\mathbf{L}^2} + \\langle \\partial_i \\mathbf{d}, \\partial_i [(\\mathbf{d} \\times \\mathbf{h}) \\times \\mathbf{h} ] \\rangle =& \\lvert \\mathbf{d} \\times \\partial_i \\mathbf{h} \\rvert^2_{\\mathbf{L}^2}+ \\langle \\partial_i \\mathbf{d} \\times \\mathbf{h}, \\mathbf{d} \\times \\partial_i \\mathbf{h} \\rangle + \\langle \\partial_i \\mathbf{d}, (\\mathbf{d} \\times \\mathbf{h})\\times \\partial_i \\mathbf{h} \\rangle \\\\\n\t\t&\\le \\lvert \\partial_i \\mathbf{d} \\rvert^2_{\\mathbf{L}^4} \\lvert \\partial_i \\mathbf{h} \\rvert^2_{\\mathbf{L}^4} + \\lvert \\partial_i \\mathbf{d} \\rvert^2_{\\mathbf{L}^2} \\lvert \\mathbf{d} \\rvert^2_{\\mathbf{L}^4} \\lvert \\mathbf{h} \\rvert_{\\mathbf{L}^\\infty} \\lvert \\partial_i \\mathbf{h} \\rvert_{\\mathbf{L}^4} \\le C \\lVert \\mathbf{d} \\rVert^2_{1} \\lvert \\mathbf{h} \\rvert^2_{\\mathbf{W}^{1,4}}.\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tHence, summing over $i$ from $1$ to $2$ imply the desired inequality \\eqref{Eq:EstimateItoStratoCorrect}.\n\t\\end{proof}\n\tWe now give the promised of the proposition.\n\t\\begin{proof}[Proof of Proposition \\ref{EST1} ]\n\t\tWithout loss of generality (Wlog) we only give a proof for $p=2(4N+1)$. Let $\\mathbf{h} \\in \\mathbf{W}^{1,4}$. Throughout this proof $C=C(\\lVert \\mathbf{h} \\rVert_{p, \\mathbf{W}^{1,4}})>0$ is a constant which may change from one term to the next one. Let $k \\in \\mathbb{N}$ be fixed and $\\tau_k$ be defined by \\eqref{STOP} . \n\t\t\n\t\tFirstly, let $\\Lambda:\\mathbf{H}^1 \\to [0,\\infty)$ be the map defined by\n\t\t\\begin{equation}\n\t\t\\Lambda(\\mathbf{d})= \\frac12 \\lvert \\mathbf{d} \\rvert^2_{\\mathbf{L}^2} + \\lvert \\nabla \\mathbf{d} \\rvert^2_{\\mathbf{L}^2} +\\frac12 \\int_\\mathcal{O} F(\\lvert \\mathbf{d}(x) \\rvert^2) dx, \\;\\; \\mathbf{d}\\in \\mathbf{H}^1.\n\t\t\\end{equation}\n\t\tBy Assumption \\ref{eqn-f} and \\cite[Lemma 8.10]{Brz+Millet_2012} the map $\\Lambda(\\cdot)$ is twice Fr\\'echet differentiable. Moreover, elementary calculations and \\eqref{Eq:MixedScalCrossProducts} imply \n\t\t\\begin{align}\n\t\t& \\Lambda^\\prime(\\mathbf{d})[G(\\mathbf{d})]= \\langle \\nabla \\mathbf{d}, \\mathbf{d}\\times \\nabla \\mathbf{h} \\rangle,\\label{Eq:PsiDerGd}\\\\\n\t\t& \\frac12 \\Lambda^\\prime (G^2(\\mathbf{d})) + \\frac12\\Lambda^{\\prime \\prime}[G(\\mathbf{d}), G(\\mathbf{d}) ]=\\frac12 \\lvert \\nabla G(\\mathbf{d}) \\rvert^2_{\\mathbf{L}^2} + \\frac12 \\langle \\nabla \\mathbf{d}, \\nabla G^2(\\mathbf{d}) \\rangle. \\label{Eq:ItoStrato-Correct}\n\t\t\\end{align}\n\t\tWe also observe that if $\\mathbf{u} \\in \\mathrm{V}$ such that $\\Div \\mathbf{u}=0$, then\n\t\t\\begin{equation}\n\t\t\\langle \\mathbf{u} \\cdot \\nabla \\mathbf{d}, f(\\mathbf{d}) \\rangle = \\frac12 \\int_\\mathcal{O} \\mathbf{u}(x)\\cdot \\nabla F(\\mathbf{d}(x)) dx=0.\\label{Eq:Divvimply}\n\t\t\\end{equation}\n\t\t\n\t\tSecondly, applying the It\\^o formula to $\\frac12 \\lvert \\v(t\\wedge \\tau_k)\\rvert^2_{\\mathrm{L}^2} + \\Lambda(\\d(t\\wedge \\tau_k)) $ and using \\eqref{tild-b-0}, \\eqref{B3}, \\eqref{G1-eq-Md}, \\eqref{Eq:PsiDerGd}, \\eqref{Eq:ItoStrato-Correct} and \\eqref{Eq:Divvimply} yield\n\t\t\\begin{equation}\\label{Eq:ResultIto}\n\t\t\\begin{split}\n\t\t&\\mathcal{E}[\\v,\\d](t\\wedge \\tau_k)-\\mathcal{E}[\\v,\\d](0) + \\int_0^{t\\wedge \\tau_k} \\mathscr{D}[\\v,\\d] (s)\\, ds - \\int_0^{t\\wedge \\tau_k} \\langle \\d(s) , f(\\d(s)) \\rangle ds-\\frac12 \\int_0^{t\\wedge \\tau_k} \\lvert \\nabla G(\\d(s)) \\rvert^2_{\\mathbf{L}^2} ds \\\\\n\t\t& = \\frac12 \\int_0^{t\\wedge \\tau_k} \\langle \\nabla \\mathbf{n}(s), \\nabla G^2(\\mathbf{n}(s)) \\rangle ds +\n\t\t\\int_0^{t\\wedge \\tau_k} \\langle \\v(s), S(\\v(s)) dW_1(s) \\rangle + \\int_0^{t\\wedge \\tau_k} \\langle \\nabla \\d(s), \\mathbf{n}(s) \\times \\nabla \\mathbf{h}\\rangle dW_2.\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tBefore proceeding further, we should observe that thanks to Assumption \\ref{eqn-f} and \\cite[Lemma 8.7]{Brz+Millet_2012} we infer that there exists exists $c>0$ independent of $k$ such that \n\t\t\\begin{equation*}\n\t\t\\frac{-a_{N+1}}{2} \\int_{\\mathcal{O}} \\lvert \\d(x)\\rvert^{2N+2} dx-c \\int_{\\mathcal{O}} \\lvert \\d(x)\\rvert^2 dx \\le \\langle -f(\\d), \\d\\rangle.\n\t\t\\end{equation*}\n\t\tHence, plugging this inequality and \\eqref{Eq:EstimateItoStratoCorrect} into \\eqref{Eq:ResultIto} implies\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t&\\mathcal{E}[\\v,\\d](t\\wedge \\tau_k)-\\mathcal{E}[\\v,\\d](0) + \\int_0^{t\\wedge \\tau_k} \\mathscr{D}[\\v,\\d] (s)\\, ds -\\frac{a_{N+1}}{2} \\int_0^{t\\wedge \\tau_k} \\lvert \\d(s) \\rvert^{2N+2}_{\\mathrm{L}^{2N+2}} ds\\\\\n\t\t& \\le \\int_0^{t\\wedge \\tau_k} \\langle \\v(s), S(\\v(s)) dW_1(s) \\rangle + \\int_0^{t\\wedge \\tau_k} \\langle \\nabla \\d(s), \\mathbf{n}(s) \\times \\nabla \\mathbf{h}\\rangle dW_2+ \\frac12 \\int_0^{t\\wedge \\tau_k} \\mathcal{E}[\\v,\\d](s)ds .\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tThirdly, by taking the supremum over $s\\in [0,t]$, raising to the power $p$, taking the mathematical expectation to both sides of the above inequality and applying the H\\\"older inequality we obtain\n\t\t\\begin{equation}\\label{Eq:PowerPofNRJ}\n\t\t\\begin{split}\n\t\t& \\mathbb{E} \\sup_{s\\in [0,t]}\\lvert \\mathcal{E}[\\v,\\d](s\\wedge \\tau_k)\\rvert^p + \\mathbb{E} \\left[ \\int_0^{t\\wedge \\tau_k}\\left(\\mathscr{D}[\\v,\\d](s) - \\frac{a_{N+1}}{2} \\lvert \\mathbf{n}(s) \\rvert^{2N+2}_{\\mathbf{L}^{2N+2}} \\right)ds \\right]^p\\\\\n\t\t& \\le C\\mathbb{E}\\lvert \\mathscr{E}[\\v,\\d] (0)\\rvert^p+Ct^{p-1}\\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\rvert \\mathcal{E}[\\v,\\mathbf{n}](s)\\lvert^p ds + C \\mathbb{E}\\sup_{s\\in [0,t]} \\biggl\\lvert \\int_0^{s\\wedge \\tau_k}\\langle \\v(s), S(\\v(s)) dW_1(s) \\biggr\\rvert^p\\\\\n\t\t& \\qquad +\n\t\tC \\mathbb{E}\\sup_{s\\in [0,t]} \\biggl\\lvert \\int_0^{s\\wedge \\tau_k}\\langle \\nabla \\d(s), \\mathbf{n}(s) \\times \\mathbf{h} \\rangle dW_2(s) \\biggr\\rvert^p.\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tFor the time being let us assume that there exists $C=C(p,\\lVert \\mathbf{h} \\rVert_{\\mathbf{W}^{1,4}})>0$ such that\n\t\t\\begin{equation}\n\t\t\\begin{split}\n\t\t& \\mathscr{M}(t\\wedge \\tau_k):=C \\mathbb{E}\\sup_{s\\in [0,t]} \\biggl\\lvert \\int_0^{s\\wedge \\tau_k}\\langle \\v(s), S(\\v(s)) dW_1(s) \\biggr\\rvert^p+\n\t\tC \\mathbb{E}\\sup_{s\\in [0,t]} \\biggl\\lvert \\int_0^{s\\wedge \\tau_k}\\langle \\nabla \\d(s), \\mathbf{n}(s) \\times \\mathbf{h} \\rangle dW_2(s) \\biggr\\rvert^p\\\\\n\t\t&\\qquad \\qquad \\le \\frac12 \\mathbb{E} \\sup_{s\\in [0,t]} \\lvert \\mathcal{E}[\\v,\\d](s\\wedge \\tau_k) \\rvert^p + C_1t^\\frac{p}2 + C_1 t^{\\frac{p-2}{2}}\\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\lvert \\mathscr{E}[\\v,\\d](s)\\rvert^p ds,\\label{Eq:EstPowerPofMart}\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tfrom which along with \\eqref{Eq:PowerPofNRJ} we infer that there exists $C=C(p,\\lVert \\mathbf{h} \\rVert_{\\mathbf{W}^{1,4}})>0$ such that\n\t\t\\begin{equation\n\t\t\\begin{split}\n\t\t& \\mathbb{E} \\sup_{s\\in [0,t]}\\lvert \\mathcal{E}[\\v,\\d](s\\wedge \\tau_k)\\rvert^p + 2 \\mathbb{E} \\left[ \\int_0^{T\\wedge \\tau_k}\\left(\\mathscr{D}[\\v,\\d](s) - \\frac{a_{N+1}}{2} \\lvert \\mathbf{n}(s) \\rvert^{2N+2}_{\\mathbf{L}^{2N+2}} \\right)ds \\right]^p\\\\\n\t\t& \\le C \\mathbb{E}\\lvert \\mathscr{E}[\\v,\\d] (0)\\rvert^p+ Ct^\\frac{p}2 + C(t^{p-1}+ t^{\\frac{p-2}{2}}) \\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\rvert \\mathcal{E}[\\v,\\mathbf{n}](s)\\lvert^p ds.\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tWe then apply the Gronwall lemma and obtain the desired \\eqref{Eq:ESTofVDinWeakerNorm}.\\\\ \n\t\tThus, it remains to prove \\eqref{Eq:EstPowerPofMart}. For this purpose, by applying the Burkholder-Davis-Gundy (BDG), the H\\\"older and Young inequalities, and Assumption \\ref{HYPO-ST} (mainly \\eqref{Eq:Hypo-ST}) we infer that \n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\mathscr{M}(t\\wedge \\tau_k)& \\le C_4 \\mathbb{E} \\left[\\int_0^{t\\wedge \\tau_k} \\lvert \\v(s) \\rvert^2_{\\mathrm{L}^2} \\lvert S(\\v(s)) \\rvert^2_{\\mathcal{T}_2} \\right]^\\frac{p}2+C_4 \\mathbb{E} \\left[\\int_0^{t\\wedge \\tau_k} \\lvert \\nabla \\d(s) \\rvert^2_{\\mathrm{L}^2} \\lvert \\d(s) \\times \\nabla \\mathbf{h} \\rvert^2_{\\mathbf{L}^2} \\right]^\\frac{p}2\\\\\n\t\t& \\le C_4 t^\\frac{p-2}{2} \\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\rvert \\mathcal{E}[\\v,\\d](s)\\rvert^{\\frac{p}{2}} [\\lvert S(\\v(s)) \\rvert^2_{\\mathcal{T}_2} + \\lvert \\d(s) \\times \\nabla \\mathbf{h} \\rvert^2_{\\mathbf{L}^2} ]^\\frac{p}{2} ds\\\\\n\t\t&\\le \\frac12 \\mathbb{E} \\sup_{s\\in [0,t]} \\mathbb{E} \\rvert \\mathcal{E}[\\v,\\d](s\\wedge \\tau_k)\\rvert^{p} +C_5 t^{p-2} \\mathbb{E} \\int_0^{t\\wedge \\tau_k} [1+ \\lvert\\v(s)\\rvert^2_{\\mathrm{L}^2} + \\lvert \\d(s) \\rvert^2_{\\mathbf{L}^4} \\lvert \\nabla \\mathbf{h} \\rvert^2_{\\mathbf{L}^4} ]^p ds.\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tThe last line and $\\mathbf{H}^1\\hookrightarrow \\mathbf{L}^4$ imply \\eqref{Eq:EstPowerPofMart}. This completes the proof of Proposition \\ref{EST1}.\n\t\\end{proof}\n\t\\subsection{Proof of Proposition \\ref{STRONGER-NORM}}\n\tBefore giving the promised proof we firstly state and prove two important lemmata. The first one is inspired by \\cite[Lemma 6.3]{ZB+MO}.\n\t\\begin{lem}\\label{Lem:Abstract-Lem-L(x)}\n\t\tLet $V_1, H_1, \\tilde{V}_1$ be there separable Hilbert spaces such that the embeddings $V_1\\hookrightarrow H_1\\hookrightarrow \\tilde{V}_1$ are dense and continuous. Let $\\mathcal{A}: V_1 \\to \\tilde{V}_1$ be a bounded linear map and $\\mathfrak{f}: [0,T] \\to H_1$ and $g: [0,T] \\to V_1$ measurable and progressively measurable respectively such that\n\t\t\\begin{equation}\\label{Eq:Integrabilitoffandg}\n\t\t\\mathbb{E}\\int_0^T[ \\lvert \\mathfrak{f}(t) \\rvert^2_{\\tilde{V}_1} + \\lvert g(t) \\rvert^2_{H_1} ]dt <\\infty.\n\t\t\\end{equation}\nLet $x: [0,T]\\times \\Omega \\to V_1$ be a progressively measurable and $H_1$-continuous process such that \n\t\\begin{align}\n\t&\t\\mathbb{E}\\int_0^T \\lvert x(s) \\rvert^2_{V_1}ds<\\infty,\\label{Eq:Integrabilityofx}\\\\\n&\tx(t) =x(0) + \\int_0^t \\mathcal{A}x(s)\\, ds + \\int_0^t \\mathfrak{f}(s)\\, ds + \\int_0^t g(s) dW_2(s)\\, \t\\text{ for all $t$ $\\mathbb{P}$-a.s..} \t\\label{Eq:Equationforx}\n\t\\end{align}\n\tNow, let $V_2, H_2$ be two separable Hilbert spaces and $V_2^\\ast$ the dual of $V_2$. We identify $H_2$ with its dual and we assume that the embeddings $V_2\\hookrightarrow H_2\\hookrightarrow V_2^\\ast $ are continuous and dense. Let $\\mathcal{B}: V_2 \\to V_2^\\ast$ be a bounded linear map. Let $L: \\tilde{V}_1 \\to V_2^\\ast$ be a twice Fr\\'echet differentiable map such that:\n\t\\begin{enumerate}\n\t\t\\item $L(V_1)\\subset V_2$ and $L(H_1)\\subset H_2$,\n\t\t\\item there exists $\\mathcal{H}: V_1 \\to H_2$ such that for every $z\\in V_1$\n\t\t\\begin{equation}\\label{Eq:LprimeAz=Bz+H(z)}\n\t\tL^\\prime(z)[\\mathcal{A}z]=\\mathcal{B}L(z) +\\mathcal{H}(z).\n\t\t\\end{equation}\n\t\t\\item The map $L^{\\prime\\prime}$ is bounded on balls.\n\t\\end{enumerate}\n\t\n\tThen for every $t\\in [0,T]$, $\\mathbb{P}$-a.s. the following identity holds in $V_2^\\ast$\n\t\\begin{equation}\\label{Eq:IdentityforL(x)inDualV1}\n\\begin{split}\n\tL(x(t)) =L(x(0)) +\\int_0^t\\mathcal{B}L(x(s))\\, ds + \\int_0^t\\Bigl( L^\\prime(x(s))[\\mathfrak{f}(s)] +\\mathcal{H}(x(s)) \\Bigr)\\, ds \\\\+ \\frac12 \\int_0^t L^{\\prime\\prime}(x(s))[g(s), g(s)] ds+ \\int_0^t L^\\prime(x(s))[g(s)]dW_2(s).\n\\end{split}\n\t\\end{equation}\n\t\\end{lem}\n\\begin{proof}\n\tLet $t\\in [0,T]$ and $\\varphi \\in V_2$. Let $z\\ni L_\\varphi:H_1 \\mapsto L_\\varphi(z):= {}_{V_2^\\ast}\\langle L(z), \\varphi \\rangle_{V_2}\\in \\mathbb{R}$.\n\tBy the assumptions on $L$, $L_\\varphi$ is twice Fr\\'echet differentiable, $L_\\varphi$ and $L^\\prime_\\varphi$ are continuous on $H_1$, $L_\\varphi$, $L_\\varphi^\\prime$ and $L_\\varphi^{\\prime\\prime}$ are locally bounded. Hence, by It\\^o formula, see \\cite[Theorem 3.2]{Pardoux}, we infer that $\\mathbb{P}$-a.s.\n\\begin{equation*}\n\\begin{split}\nL_\\varphi(x(t))=L_\\varphi(x(0)) + \\int_0^t {}_{V_2^\\ast}\\langle L^\\prime(x(s))[\\mathcal{A}(x(s)) +\\mathfrak{f}(s) ], \\varphi \\rangle_{V_2} + \\int_0^t {}_{H_2}\\langle L^\\prime(x(s)) [g(s)],\\varphi \\rangle_{H_2} dW_2(s)\\\\+ \\frac12\\int_0^t {}_{H_2}\\langle L^{\\prime\\prime}(x(s))[g(s), g(s)], \\varphi\\rangle_{H_2} ds\n\\end{split}\n\\end{equation*}\nUsing \\eqref{Eq:LprimeAz=Bz+H(z)} yields\n\\begin{equation*}\n\\begin{split}\nL_\\varphi(x(t))=L_\\varphi(x(0)) + \\int_0^t {}_{V_2^\\ast}\\langle \\mathcal{B}(x(s)), \\varphi \\rangle_{V_2} ds +\\int_0^t {}_{V_2^\\ast}\\langle L^\\prime(x(s))[ \\mathfrak{f}(s)]+H(x(s)) , \\varphi \\rangle_{V_2}ds \\\\+ \\frac12\\int_0^t {}_{H_2}\\langle L^{\\prime\\prime}(x(s))[g(s), g(s)], \\varphi\\rangle_{H_2} ds +\n\\int_0^t {}_{H_2}\\langle L^\\prime(x(s)) [g(s)],\\varphi \\rangle_{H_2} dW_2(s) .\n\\end{split}\n\\end{equation*}\nThis completes the proof of \\eqref{Eq:IdentityforL(x)inDualV1}.\n\\end{proof}\n\t\\begin{lem}\\label{Lem:IdentityforDd+f(d)}\n\tLet $\\mathbf{h}\\in \\mathbf{H}^2$ and $\\{y(t): t\\in [0,\\tilde{\\tau}_\\infty)\\}$ be the local process defined by\n\t\\begin{equation}\\label{Eq:Def-Dd+f(d)}\n\ty(t):=\n\t \\mathrm{A}_1 \\mathbf{n}(t) - f(\\mathbf{n}(t)), \\;\\text{ } t\\in [0,\\tilde{\\tau}_\\infty).\n\t\\end{equation}\n\tThen, for all $t\\in [0,T]$, $k \\in \\mathbb{N}$, $\\mathbb{P}$-a.s. the following equation holds in $(\\mathbf{H}^1)^\\ast$\n\t\\begin{equation}\\label{Eq:IdentityforDd+f(d)}\n\\begin{split}\n&\ty(t\\wedge \\tau_k)+ \\int_0^{t\\wedge \\tau_k} \\Bigl(\\mathrm{A}_1 y(r) + (\\mathrm{A}_1-f^\\prime(\\mathbf{n}(r)))[\\mathbf{v}(r) \\cdot \\nabla \\mathbf{n}(r)] \\Bigr)\\, dr -\\int_0^{t\\wedge \\tau_k} (\\mathrm{A}_1-f^\\prime(\\mathbf{n}(r)))[G(\\mathbf{n}(r))] dW_2\\\\\n&= y(0) +\\frac12 \\int_0^{t\\wedge \\tau_k} \\Bigl(2f^\\prime(\\mathbf{n}(r))y(r) +(\\mathrm{A}_1-f^\\prime(\\mathbf{n}(r)))[G^2(\\d(r))] - f^{\\prime \\prime}(\\d(r))[G(\\d(r)), G(\\d(r))] \\Bigr)\\, dr.\n\\end{split}\n\t\\end{equation}\n\t\\end{lem}\n\\begin{proof}[Proof of Lemma \\ref{Lem:IdentityforDd+f(d)}]\n\tLet $\\mathbf{h}\\in \\mathbf{H}^2$. \n\tLet us put $V_1=D((I+\\mathrm{A}_1)^\\frac32)$, $H_1=D(\\mathrm{A}_1)$, $\\tilde{V}_1=\\mathbf{H}^1$ and $\\mathcal{A}=\\mathcal{B}=-\\mathrm{A}_1$. We also set $V_2=\\mathbf{H}^1$, $H_2=\\mathbf{L}^2$, $V_2^\\ast=(\\mathbf{H}^1)^\\ast$. The map\n\t$L: \\mathbf{H}^1 \\ni z \\mapsto L(z):= \\mathrm{A}_1z-f(z) \\in (\\mathbf{H}^1)^\\ast$satisfies the assumptions of Lemma \\ref{Lem:Abstract-Lem-L(x)}. In particular, if we set $\\mathcal{H}(z)= \\mathrm{A}_1f(z) -f^\\prime(z)[\\mathrm{A}_1z], \\,z \\in V_1 $, then \n\t$$ L^\\prime(z)[\\mathrm{A}_1z]= \\mathrm{A}_1[ \\mathrm{A}_1 z - f(z)] +\\mathrm{A}_1f(z) -f^\\prime(z)[\\mathrm{A}_1z] = \\mathrm{A}_1 L(z) +\\mathcal{H}(z)\\in (\\mathbf{H}^1)^\\ast. $$\n Now, let $k\\in \\mathbb{N}$ and\n \\begin{align}\n &\\mathfrak{f}= \\mathds{1}_{[0,\\tau_k)} [ -\\mathbf{v} \\cdot \\nabla \\mathbf{n} + f(\\mathbf{n}) +\\frac12G^2(\\mathbf{n}) ],\\label{Eq:DefFrakf}\\\\\n &g= \\mathds{1}_{[0,\\tau_k)} G(\\mathbf{n}).\\label{Eq:Defsmallg}\n \\end{align}\nBy Lemmata \\ref{Local-LIP-Lem-2} and\\ref{Local-LIP-Lem-3}, and the definition of $\\tau_k$ we infer that there exists $C>0$ such that\n \\begin{align}\n& \\mathbb{E} \\int_0^{t} \\lvert \\mathfrak{f}(r) \\rvert^2_{\\mathbf{H}^1}\\le \\mathbb{E}\\int_0^{t\\wedge \\tau_k} \\Bigl(\\lvert \\mathbf{v}(r) \\cdot \\nabla \\mathbf{n}(r)\\rvert^2_{\\mathbf{H}^1} +\\lvert f(\\mathbf{n}(r))\\rvert^2_{\\mathbf{H}^1}+\\lvert \\frac12 (\\mathbf{n}(r) \\times \\mathbf{h}) \\times \\mathbf{h} \\rvert^2_{\\mathbf{H}^1}\\Bigr)\\, dr\\le C,\\label{Eq:Proof of IdDd+f(d)-1}\\\\\n& \\mathbb{E}\\int_0^{t\\wedge \\tau_k} \\lvert G(\\mathbf{n}(r)) \\rvert^2_{\\mathbf{H}^2} dr = \t\\mathbb{E}\\int_0^{t\\wedge \\tau_k} \\lvert \\mathbf{n}(r) \\times \\mathbf{h} \\rvert^2_{\\mathbf{H}^2} dr \\le C.\\label{Eq:Proof of IdDd+f(d)-2}\n \\end{align}\n These mean that $\\mathfrak{f}$ and $g$ satisfy \\eqref{Eq:Integrabilitoffandg}. Because the local strong solution $\\mathbf{y}=(\\mathbf{v},\\mathbf{n})$ of \\eqref{ABSTRACT-LC} satisfies \\eqref{eq-locsol_01-a}, the process $x(t)=\\mathbf{n}(t\\wedge \\tau_k)$ satisfies \\eqref{Eq:Integrabilityofx} and \\eqref{Eq:Equationforx} with $\\mathfrak{f}$ and $g$ as defined above. By setting\n $y(t\\wedge \\tau_k)=L(\\mathbf{n}(t\\wedge \\tau_k)), t\\ge 0,$ and applying Lemma \\ref{Lem:Abstract-Lem-L(x)} we obtain\n \\begin{equation}\\label{Eq:AbsFormofAd+f(d)}\n \\begin{split}\n y(t\\wedge \\tau_k)=y(0) +\\int_0^{t\\wedge \\tau_k} \\Bigl( -\\mathrm{A}_1y(r) + L^\\prime(\\mathbf{n}(r)) [\\mathfrak{f}(r)] + \\mathcal{H}(\\boldsymbol{\\nu}(r)) \\Bigr)\\, dr \\\\+ \\frac12\\int_0^{t\\wedge \\tau_k} L^{\\prime \\prime}(\\mathbf{n}(r))[g(r), g(r)] dr + \\int_0^{t\\wedge \\tau_k} L^\\prime(\\mathbf{n}(r)) [g(r) ]dW_2(r).\n \\end{split}\n \\end{equation}\nWe complete the proof of the lemma by taking into account the last line and the following identity \n\\begin{equation}\\label{Eq:1stDer+2ndDerofL}\n L^\\prime(z)=\\mathrm{A}_1 -f^\\prime(z) \\text{ and } L^{\\prime\\prime } (z)= - f^{\\prime\\prime}(z) \\text{ for every } z\\in \\mathbf{H}^1.\n\\end{equation}\n \\end{proof}\n\n\n\tWe now give the promised proof of Proposition \\ref{STRONGER-NORM}.\n\t\\begin{proof}[Proof of Proposition \\ref{STRONGER-NORM}]\n\t\tThroughout, $L(z)=\\mathrm{A}_1 z-f(z)$ and $\\mathcal{H}(z)= \\mathrm{A}_1f(z) - f^\\prime(z)[\\mathrm{A}_1 z]$ be defined as in the proof of Lemma \\ref{Lem:IdentityforDd+f(d)}. \n\tKeeping in mind the notations of \\ref{Lem:IdentityforDd+f(d)}, in particular \\eqref{Eq:DefFrakf}, \\eqref{Eq:Defsmallg} and \\eqref{Eq:1stDer+2ndDerofL},\n\t\twe set\n\t\t\\begin{equation}\n\t\t\\mathfrak{v}= -\\mathds{1}_{[0,\\tau_k]} \\mathrm{A}_1 L(\\mathbf{n}) + L^\\prime(\\mathbf{n})[\\mathfrak{f} ] +\\mathds{1}_{[0,\\tau_k]}\\mathcal{H}(\\mathbf{n}) +\\frac12 L^{\\prime\\prime}(\\mathbf{n})[g, g], \\, k \\in \\mathbb{N}.\n\t\t\\end{equation}\nThen, for all $k \\in \\mathbb{N}$ and $F\\in L^2(\\Omega\\times[0,t]; \\mathbf{L}^2)$, $t\\ge 0$, \n\t\t\t\\begin{equation}\\label{Eq:Proof of IdDd+f(d)-3-b}\n\t\t\\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\lvert f^\\prime(\\mathbf{n}(r))[F(r)] \\rvert^2_{\\mathbf{L}^2} dr \\le c_1^2(1+k^2) \\mathbb{E}\\int_0^{t\\wedge \\tau_k } \\lvert F(r) \\rvert^2_{\\mathbf{L}^2}dr.\n\t\t\\end{equation}\n\tIn fact, from Remark \\ref{REM-H2}\\eqref{Itemi:REM-H2}, the embedding $\\mathbf{H}^2\\hookrightarrow \\mathbf{L}^\\infty$ and \\eqref{Eq:Proof of IdDd+f(d)-1} we infer that\n\t\t\\begin{align*}\n\t\t\\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\lvert f^\\prime(\\mathbf{n}(r))[F(r)] \\rvert^2_{\\mathbf{L}^2}\\le& \\mathbb{E}\\int_0^{t\\wedge \\tau_k} \\int_\\mathcal{O} \\lvert f^\\prime(\\mathbf{n}(r,x))[F(r,x)]\\rvert^2 dxdr\\\\\n\t\t& \\le c_1^2 \\mathbb{E} \\int_0^{t\\wedge \\tau_k} (1 + \\lvert \\mathbf{n}(r,x) \\rvert^{4N})\\lvert F(r,x)\\rvert^2 dxdr\\\\\n\t\t&\\le c_1^2 \\mathbb{E}\\biggl([1+\\sup_{t\\in [0,T]} \\lvert \\mathbf{n}(t\\wedge \\tau_k) \\rvert^{4N}_{\\mathrm{L}^\\infty}]\\int_0^{\\tau_k} \\lvert F(r) \\rvert^2_{\\mathbf{L}^2} dr \\biggr) \\\\\n\t\t&\t\\le c_1^2(1+k^{2N}) \\mathbb{E}\\int_0^{t\\wedge \\tau_k } \\lvert F(r) \\rvert^2_{\\mathbf{L}^2}dr.\n\t\t\\end{align*}\n\t\tIn a similar way, we can prove that for all $k \\in \\mathbb{N}$ and $g=1_{[0,\\tau_k]} \\mathbf{n}\\times \\mathbf{h} $\n\t\t\\begin{equation}\\label{Eq:Proof of IdDd+f(d)-4}\n\t\t\\mathbb{E} \\int_0^{t\\wedge \\tau_k} \\lvert f^{\\prime\\prime}(\\mathbf{n}(r))[g(r) , g(r) ]\\rvert^2_{\\mathbf{L}^2} \\le c^2_2 \\lVert \\mathbf{h} \\rVert^4_{2}(1+k^{2N} )\\mathbb{E}\\int_0^{t\\wedge \\tau_k } \\lvert \\mathbf{n}(r) \\rvert^2_{\\mathbf{L}^2}dr .\n\t\t\\end{equation}\n\tFrom the continuity of the linear map $\\mathrm{A}_1:\\mathbf{H}^1 \\to (\\mathbf{H}^1)^\\ast $, the embedding $\\mathbf{H}^1\\hookrightarrow \\mathbf{L}^2$, \\eqref{Eq:Proof of IdDd+f(d)-3-b}, \\eqref{Eq:Proof of IdDd+f(d)-4} along with \\eqref{Eq:Proof of IdDd+f(d)-1} and \\eqref{Eq:Proof of IdDd+f(d)-2} we infer that there exits $C=C(k)>0$ such that\n\t\t\\begin{equation}\\label{Eq:DriftsatisfiesPardouxConditions}\n\\mathbb{E}\\int_0^{t} \\lvert \\mathfrak{v}(r) \\rvert^2_{(\\mathbf{H}^1)^\\ast} \\le C \\mathbb{E}\\int_0^{t\\wedge \\tau_k}\\Bigl( \\lvert L(\\mathbf{n}(r)) \\rvert^2_{\\mathrm{L}^2} + \\lvert \\mathfrak{f}(r) \\rvert^2_{\\mathbf{H}^1} +\\lvert f(\\mathbf{n}(r) ) \\rvert^2_{\\mathbf{H}^1}\n+ \\lvert f^{\\prime \\prime}(\\mathbf{n}(r))[g(r), g(r) ] \\rvert^2_{\\mathbf{L}^2} \\Bigr)dr <\\infty.\n\t\t\\end{equation}\n\t\t\n\t\tNext, let $\\{ y(t): t\\in [0,\\tilde{\\tau}_\\infty) \\}$ be the process defined in \\eqref{Eq:Def-Dd+f(d)}. The process $\\{ y(t\\wedge \\tau_k): t \\ge 0 \\}$ is an $\\mathbf{L}^2$-valued process and satisfies the equivalent equations \\eqref{Eq:IdentityforDd+f(d)} and \\eqref{Eq:AbsFormofAd+f(d)}. Hence, by \\eqref{Eq:DriftsatisfiesPardouxConditions} we can apply the It\\^o formula to $ \\frac12\\lvert y(t\\wedge \\tau_k)\\rvert^2_{\\mathbf{L}^2}=\\frac12\\lvert \\mathrm{A}_1 \\mathbf{d} -f(\\mathbf{d}) \\rvert^2_{\\mathbf{L}^2}=\\Psi_1(\\mathbf{d})$ (see \\cite[Theorem 3.2]{Pardoux}), and use the fact\n\t\t\\begin{equation*}\n\t\t-(\\mathrm{A}_1-f^\\prime(\\mathbf{n}))[\\mathbf{v} \\cdot \\nabla \\mathbf{n}]+ (f^\\prime(\\mathbf{n})y - (\\mathrm{A}_1-f^\\prime(\\mathbf{n}))[\\frac12G^2(\\d)] - \\frac12f^{\\prime \\prime}(\\d)[G(\\d), G(\\d)] \\in \\mathbf{L}^2,\n\t\t\\end{equation*}\n\tto infer that for all $k \\in \\mathbb{N}$ and $t\\ge 0$\n\t\t\\begin{equation} \\label{Eq:Psi(d)Formnula-1}\n\t\t\\begin{split}\n\t&\t\\Psi_1(\\mathbf{n}(t\\wedge \\tau_k)) + \\int_0^{t\\wedge \\tau_k} \\lvert \\nabla (\\mathrm{A}_1\\mathbf{n}(r) -f(\\mathbf{n}(r)) ) \\rvert^2_{\\mathbf{L}^2} dr~+\\int_0^{t\\wedge \\tau_k}\\Psi^\\prime_1(\\d(r))[ \\v(r)\\cdot \\nabla \\d(r)]dr\\\\\n\t&\t=\n\t\t\\Psi_1(\\mathbf{n}_0) +\\int_0^{t\\wedge \\tau_k}\\Psi^\\prime_1(\\d(r))[ \\frac12 G^2(\\d(r))]dr + \\frac12 \\int_0^{t\\wedge \\tau_k} \\Psi_1^{\\prime\\prime}(\\mathbf{n}(r))[G(\\mathbf{n}(r)), G(\\mathbf{n}(r)) ] dr\\\\\n\t&\t- \\int_0^{t\\wedge \\tau_k} \\langle\\mathrm{A}_1\\mathbf{n}(r) -f(\\mathbf{n}(r)), f^\\prime(\\mathbf{n}(r))[\\mathrm{A}_1\\mathbf{n}(r) -f(\\mathbf{n}(r))] \\rangle dr+ \\int_0^{t\\wedge \\tau_k} \\Psi_1(\\mathbf{n}(r)) [G(\\mathbf{n}(r))]dW_2(r).\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tIn preparation of our next step, let us set\n\t\t\\begin{equation*}\n\t\t\\mathrm{M}(t\\wedge \\tau_k)= \\int_0^{t\\wedge \\tau_k} \\Phi(s) \\Psi_2(\\v(s)) \\circ S(\\v(s)) dW_1(s) + \\int_0^{t\\wedge \\tau_k} \\Phi(s) \\Psi_1(\\d(s))[G(\\d(s))] dW_2(s),\\, k \\in \\mathbb{N}, t\\ge 0. \n\t\t\\end{equation*}\n\tWith \\eqref{Eq:Psi(d)Formnula-1} and the definitions of $\\Psi_2, \\Psi$ and $\\Phi$ (see \\eqref{Eq:DefPsi2forv}, \\eqref{Eq:DefFunctionForIto} and \\eqref{Eq:WeightToCancelbadterms}) in mind, we\tapply the It\\^o formula to $\\Upsilon(t\\wedge \\tau_k) =\\Phi(t\\wedge \\tau_k) \\Psi(\\mathbf{u},\\mathbf{n})(t\\wedge \\tau_k) $ and obtain that for all $k \\in \\mathbb{N}$ and $t\\ge 0$\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\Upsilon(t\\wedge \\tau_k) -\\Upsilon(0)=& \\mathrm{M}(t\\wedge \\tau_k)+ \\int_0^{t\\wedge \\tau_k} \\Phi(s) \\Psi^\\prime_2(\\v(s)) [-B(\\v(s),\\v(s ) ) -M(\\d(s), \\d(s)) ] ds\\\\\n\t\t&+\\int_0^{t\\wedge \\tau_k} \\Phi(s) \\Psi^\\prime_1(\\d(s))[ -\\v(s)\\cdot \\nabla \\d(s) +\\frac12 G^2(\\d(s)) ]ds\\\\\n\t\t& +\\frac12 \\int_0^{t\\wedge \\tau_k} \\Phi(s) \\Psi_1^{\\prime\\prime}(\\d(s))[G(\\d(s)), G(\\d(s))] ds+\\int_0^{t\\wedge \\tau_k} \\frac{d}{ds}\\Phi(s) \\Psi(\\v(s), \\d(s)))\\, ds\\\\\n\t\t&- \\int_0^{t\\wedge \\tau_k} \\Phi(s) \\langle\\mathrm{A}_1\\mathbf{n}(s) -f(\\mathbf{n}(s)), f^\\prime(\\mathbf{n}(s))[\\mathrm{A}_1\\mathbf{n}(s) -f(\\mathbf{n}(s))] \\rangle ds\\\\\n\t\t&\t-\\int_0^{t\\wedge \\tau_k}\\Phi(s)\\left(\\lvert \\nabla (\\mathrm{A}_1\\mathbf{n}(s))-f(\\mathbf{n}(s)) \\rvert^2_{\\mathbf{L}^2}+ \\lvert \\mathrm{A} \\v(s) \\rvert^2 - \\lvert S(\\v(s)) \\rvert^2_{\\mathcal{T}_2(\\mathrm{K}_1; \\mathrm{V})} \\right)ds.\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tBy Assumption \\ref{HYPO-ST}, \\eqref{Eq:Hypo-ST-Rem}, Lemma \\ref{Lem:FirstDerAppliedtoVdotNabalaD}-\\ref{Lem:CouplingTermIto} and the facts $\\lvert \\Phi \\rvert\\le 1$ and\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\frac{d}{ds}\\Phi(s) \\Psi(\\v(s), \\d(s))) = &-\\left[ ((\\kappa_1 +\\kappa_4) \\lVert \\d(s) \\rVert^2_2 + \\kappa_2) (1+ \\lVert \\d(s) \\rVert^2_1) + \\kappa_3 \\lvert \\v(s) \\rvert^2_{\\mathrm{L}^2} \\lvert \\nabla \\v(s) \\rvert^2_{\\mathrm{L}^2} \\right] \\\\\n\t\t& \\qquad \\qquad \\times \\Phi(s)\\Psi(\\v(s), \\d(s)),\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\twe infer that there exists $\\kappa_7>0$ such that for all $k \\in \\mathbb{N}$ and $t\\ge 0$\n\t\t\\begin{equation}\\label{Eq:ItoresultStrongnormMart}\n\t\t\\begin{split}\n\t\t&\\mathbb{E} \\sup_{s\\in [0,t]} \\Upsilon(s\\wedge \\tau_k) + \\int_0^{t\\wedge \\tau_k} \\Phi(s) \\left[\\frac14 \\lvert \\mathrm{A} \\v(s) \\rvert^2_{\\mathrm{L}^2} + \\frac12 \\lvert\\nabla (\\mathrm{A}_1 \\d(s) +f(\\d(s)) ) \\rvert^2_{\\mathrm{L}^2} \\right] ds \\\\\n\t\t& \\le \\Upsilon(0) + \\kappa_7 T+ \\kappa_7 \\mathbb{E} \\int_0^{t\\wedge\\tau_k} \\Upsilon(s)\\, ds + \\kappa_5\\mathbb{E} \\int_0^{t\\wedge \\tau_k} (1+ \\lVert \\d(s) \\rVert^{4N}_1) \\lVert \\d(s) \\rVert^2_{2} ds + \\mathbb{E} \\sup_{s\\in [0,t]}\\lvert \\mathrm{M}(s\\wedge \\tau_k) \\rvert.\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\t\\noindent Next, by applying the BDG inequality, taking into account Assumption \\ref{HYPO-ST}, \\eqref{Eq:Psi1BDGNeeded} and the fact $\\lvert \\Phi \\rvert\\le 1 $ we infer that there exists $\\kappa_8>0$ such that for all $k \\in \\mathbb{N}$ and $t\\ge 0$\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t&\t\\mathbb{E} \\sup_{s\\in [0,t]}\\lvert \\mathrm{M}(s\\wedge \\tau_k) \\rvert \\\\\n\t& \\le \\kappa_8\\mathbb{E} \\left(\\int_0^{t\\wedge \\tau_k} \\Phi(s) \\Psi_2(s)[1+ \\Psi_2(s)] \\Phi(s)\\, ds \\right)^\\frac12 + \\kappa_8\n\t\t\\mathbb{E} \\left(\\int_0^{t\\wedge \\tau_k} \\Phi^2(s) \\lvert \\Psi_1^\\prime(\\d(s))[G(\\d(s))]\\rvert^2 ds \\right)^\\frac12 \\\\\n\t\t& \\le \\frac18 \\mathbb{E} \\sup_{s\\in [0,t]} [\\Phi(s)\\Psi_2(\\v(s)) ]+ \\kappa_8 T +\\kappa_8 \\mathbb{E} \\left[\\int_0^{t\\wedge \\tau_k} [\\Phi(s)\\Psi_1(\\d(s))]^2 ds \\right]^\\frac12\\\\\n\t\t& \\qquad \\qquad + \\kappa_8 \\mathbb{E} \\left[\\int_0^{t\\wedge \\tau_k} \\left(1 + \\lVert \\d(s) \\rVert^{8N+4}_1 + [\\lVert \\d(s) \\rVert_1 \\lVert \\d(s)\\rVert_2]^2 \\right)\\, ds \\right]^\\frac12 +\\kappa_8 \\mathbb{E}\\int_0^{t\\wedge \\tau_k} \\Phi(s) \\Psi_2(\\v(s))ds \\\\\n\t\t&\\le \\frac14 \\mathbb{E} \\sup_{s\\in [0,t]} \\Upsilon(s)+ \\kappa_8 T + \\kappa_8 \\mathbb{E}\\int_0^{t\\wedge \\tau_k} \\Upsilon(s)\\, ds + \\kappa_8 \\mathbb{E} \\left[\\int_0^{t\\wedge \\tau_k} \\left(1 + \\lVert \\d(s) \\rVert^{8N+2}_1 + [\\lVert \\d(s) \\rVert_1 \\lVert \\d(s)\\rVert_2]^2 \\right)\\, ds \\right]^\\frac12.\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tUsing the last inequality and absorbing the term $\\frac14 \\mathbb{E} \\sup_{s\\in [0,t]} \\Upsilon(s)$ in the LHS of \\eqref{Eq:ItoresultStrongnormMart}, and applying Gronwall inequality imply that there exist an increasing function $\\psi:[0,\\infty) \\to (0,\\infty)$ and a constant $\\kappa_9>0$ such that for all $k \\in \\mathbb{N}$ and $T\\ge 0$ \n\t\t\\begin{equation}\\label{Eq:FinalStepEstinStrongNorm}\n\t\t\\begin{split}\n\t\t\\mathbb{E} \\sup_{s\\in [0,T]} \\Upsilon(s\\wedge \\tau_k) +\\mathbb{E} \\int_0^{T\\wedge \\tau_k} \\Phi(s) \\left[\\frac14 \\lvert \\mathrm{A} \\v(s) \\rvert^2_{\\mathrm{L}^2} + \\frac12 \\lvert\\nabla (\\mathrm{A}_1 \\d(s) +f(\\d(s)) ) \\rvert^2_{\\mathrm{L}^2} \\right] ds\\\\\n\t\t\\le\n\t\t\\psi(T) \\biggl(1+ \\Upsilon(0) + \\mathbb{E} \\left[\\int_0^{T\\wedge \\tau_k} \\left(1 + \\lVert \\d(s) \\rVert^{8N+2}_1 + [\\lVert \\d(s) \\rVert_1 \\lVert \\d(s)\\rVert_2]^2 \\right)\\, ds \\right]^\\frac12\\\\\n\t\t+\\mathbb{E} \\int_0^{T\\wedge \\tau_k} (1+ \\lVert \\d(s) \\rVert^{4N}_1) \\lVert \\d(s) \\rVert^2_{2} ds \\biggr)\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tUsing the estimate \\eqref{Eq:ESTofVDinWeakerNorm} we easily conclude that there exists a constant $\\tilde{\\kappa}_0>0$ such that\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t& \\mathbb{E} \\left[\\int_0^{t\\wedge \\tau_k} \\left(1 + \\lVert \\d(s) \\rVert^{8N+2}_1 + [\\lVert \\d(s) \\rVert_1 \\lVert \\d(s)\\rVert_2]^2 \\right)\\, ds \\right]^\\frac12\n\t\t+\\mathbb{E} \\int_0^{t\\wedge \\tau_k} (1+ \\lVert \\d(s) \\rVert^{4N}_1) \\lVert \\d(s) \\rVert^2_{2} ds \\\\\n\t\t& \\qquad \\qquad \\le \\tilde{\\kappa}_0 (\\mathfrak{C}_0+1),\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\twhich along with \\eqref{Eq:FinalStepEstinStrongNorm} complete the proof of Proposition \\ref{STRONGER-NORM}\n\t\\end{proof}\n\n\n\n\n\n\t\\section{Strong solution for an abstract stochastic equation}\\label{ABST-STRONG}\nBy a fixed point method we prove in this section general results about the existence and uniqueness of maximal local solution to stochastic evolution equations (SEEs) with \n\tLipschitz coefficients. \n\t\\subsection{Notations and Preliminary}\\label{abstract-framework}\n\tLet $V$, $E$ and $H$ be separable Banach spaces such that\n\t$E\\hookrightarrow V$. We denote the norm in $V$ by $\\Vert\n\t\\cdot \\Vert$ and for $a,b\\in [0,\\infty)$ with $a0$\n\t\tsuch that \t\tfor all $x, y\\in E$.\n\t\t\\begin{equation}\\label{eqn-local Lipschitz-F}\n\t\t\\begin{split}\n\t\t\\vert F(y)-F(x) \\vert_H \\leq C \\sum_{i=1}^N \\Big[ \\Vert y-x\\Vert \\Vert\n\t\ty\\Vert^{p_i-\\alpha_i} \\vert y\\vert_E^{\\alpha_i} + \\vert y-x\\vert_E^{\\alpha_i}\n\t\t\\Vert y-x\\Vert^{1-\\alpha_i} \\Vert x\\Vert^{p_i}\\Big].\n\t\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\\end{assum}\n\t\\begin{assum}\\label{assum-G}\n\t\tAssume that $G: E \\to V$ such that $G(0)=0$ and there exists $k\n\t\t\\geq 1$, $\\beta \\in [0,1)$ and $C_G>0$ such that\n\t\t\\begin{equation}\\label{eqn-local Lipschitz-G}\n\t\t\\Vert G(y)-G(x) \\Vert \\leq C_G \\Big[ \\Vert y-x\\Vert \\Vert\n\t\ty\\Vert^{k-\\beta} \\vert y\\vert_E^\\beta + \\vert y-x\\vert_E^\\beta\n\t\t\\Vert y-x\\Vert^{1-\\beta} \\Vert x\\Vert^k\\Big],\n\t\t\\end{equation}\n\t\tfor all $x, y\\in E$.\n\t\\end{assum}\n\tLet $(\\Omega,\n\t\\mathcal{F}, \\mathbb{P})$ be a complete probability space equipped\n\twith a filtration $\\mathbb{F}=\\{\\mathcal{F}_t: t\\geq 0\\}$\n\tsatisfying the usual hypothesis. By $\\mathscr{M}^2(X_T)$ we denote the Banach space of all $E$-valued processes $u$ that are progressively measurable and with trajectories belonging to $X_T$ $\\mathbb{P}$-a.s., with the norm\n\t\\begin{equation}\\label{eqn-M^2X_T}\n\t\\vert u \\vert_{ \\mathscr{M}^2(X_T)} = \\left(\\mathbb{E}\\Big[ \\sup_{s \\in [0,T]}\n\t\\Vert u(s)\\Vert^2+\\int_0^T \\vert u(s) \\vert_E^2\\, ds\\Big].\\right)^\\frac12\n\t\\end{equation}\n\tLet us also formulate the following assumptions.\n\t\\begin{assum}\\label{assum-01}\n\t\tSuppose that $E\\hookrightarrow V \\hookrightarrow H$. Consider (for\n\t\tsimplicity) a one-dimensional Wiener process $W(t)$.\\\\\n\t\tAssume that $S(t)$, $t\\in [0,\\infty)$, is a family of bounded linear operators on the space $H$ such that: the following properties are satisfied. \n\t\t\\begin{trivlist}\n\t\t\t\t\\item[(i)] For every $T>0$, the linear map $$L^2(0,T;H) \\ni f \\mapsto \\{S\\ast f(t) =\\int_0^{t} S(t-r) f(r)\\, dr; t\\in [0,T] \\} \\in X_T, $$ is continuous. \n\t\t\t\t\n\\item[(ii) ]For every $T>0$, the linear map \n\t\t\t\t$$\\mathscr{M}^2(0,T; V) \\ni \\xi \\mapsto \\{ S \\diamond \\xi(t):= \\int_0^t S(t-r) \\xi(r)\\, dW(r); t \\in [0,T] \\}\\in \\mathscr{M}^2(X_T), $$ is continuous. \n\t\t\t\t\n\t\t\\item[(iii)] For every $T>0$, the linear map $$V\\ni u_0 \\mapsto \\{[0,T]\\ni t \\mapsto Su_0(t):=S(t)u_0 \\} \\in X_T,$$ is continuous.\n\t\t\\end{trivlist}\n\t\\end{assum}\n\tNow let us consider a semigroup $S(t)$, $t\\in [0,\\infty)$ as above\n\tand the abstract SEE\n\t\\begin{equation}\\label{ABS-SPDE-1}\n\tu(t)=S(t)u_0+\\int_0^t S(t-s) F(u(s))\\, ds+\\int_0^t S(t-s) G(u(s))\n\tdW(s),\\;\\; \\mbox{ for all }t>0\n\t\\end{equation}\n\t\n\twhich is a mild version of the problem\n\t\n\t\\begin{equation}\\label{ABS-SPDE-strong}\n\t\\left\\{\\begin{array}{rl} du(t)&= Au(t)\\,dt+ F\\big(u(t)\\big)\\, dt+\n\tG\\big(u(t)\\big)\n\tdW(t),\\;\\;t>0,\\\\\n\tu(0)&=u_0.\n\t\\end{array}\n\t\\right.\n\t\\end{equation}\n\t\\begin{Def}\\label{def-local solution-2}\n\t\tAssume that a $V$-valued $\\mathcal{F}_0$ measurable random variable $u_0$ is given. A local\n\t\tsolution to problem \\eqref{ABS-SPDE-strong} (with the initial time\n\t\t$0$) is a pair $(u,\\tau)$ such that\n\t\t\\begin{enumerate}\n\t\t\t\\item $\\tau$ is an accessible stopping time, \\item\n\t\t\t$u: [0,\\tau)\\times \\Omega \\to V$ is an admissible\\footnote{This\n\t\t\t\talso follows from condition (3) below.} process, \\item there\n\t\t\texists a sequence $(\\tau_m)_{m\\in \\mathbb{N}}$ of\n\t\t\t finite stopping times such that $\\tau_m \\toup \\tau$\n\t\t\t$\\mathbb{P}$-a.s. and, for every $m\\in \\mathbb{N}$ and $t\\ge 0$,\n\t\t\twe have\n\t\t\t\\begin{eqnarray}\\label{eq-locsol_00}\n\t\t\t&&\\mathbb{E}\\Big( \\sup_{s\\in [0,t\\wedge \\tau_m]} \\Vert\n\t\t\tu(s)\\Vert^2 +\\int_0^{t\\wedge \\tau_m} \\vert u(s)\\vert_E^2 \\,\n\t\t\tds\\Big)<\\infty,\n\t\t\t\\\\\n\t\t\t\\label{eq-locsol_00-b} u(t\\wedge \\tau_m)&=&S(t\\wedge\n\t\t\t\\tau_m)u_0+\\int_0^{t\\wedge \\tau_m}\n\t\t\tS(t\\wedge\\tau_m-s) F(u(s))\\, ds\\\\\n\t\t\t\\nonumber &+&\\int_0^t\n\t\t\t\\mathds{1}_{[0,\\tau_m)}S(t-s)G(u(s\\wedge\\tau_m))\\, dW(s).\n\t\t\t\\end{eqnarray}\n\t\t\\end{enumerate}\n\t\tAlong the lines of \\cite{Brz+Elw_2000}, we said that a\n\t\tlocal solution $u(t)$, $t < \\tau$ is called global iff\n\t\t$\\tau=\\infty$ $\\mathbb{P}$-a.s.\n\t\\end{Def}\n\n\tLet us first formulate the following useful result.\n\t\\begin{prop}\n\t\t\\label{prop-local solution} Assume that a pair $(u,\\tau)$ is a\n\t\tlocal solution to problem \\eqref{ABS-SPDE-strong}.Then for\n\t\tevery finite stopping time $\\sigma$, a pair $(u_{\\vert [0, \\tau\n\t\t\t\\wedge \\sigma)\\times \\Omega}, \\tau \\wedge \\sigma)$ is also a local\n\t\tmild solution to problem \\eqref{ABS-SPDE-strong}.\n\t\\end{prop}\n\t\n\t\n\t\n\tSecondly, we state the following lemma result which is a generalisation of \\cite[Lemmata III 6A and\n\t6B]{Elw_1982}.\n\n\t\\begin{lem} \\label{lem-amalgamation} \\textbf{(The Amalgamation\t\t\tLemma) }\n\\begin{trivlist}\n\\item[(1)]\nLet $\\Delta$ be a family of accessible stopping\n\t\ttimes taking values in $[0, \\infty ]$. Then a supremum of $\\Delta$, i.e., \n\t\t$ \\tau := \\sup \\, {\\Delta} $, \n\t is an accessible stopping time with values in $[0, \\infty ]$ and\n\t\tthere exists an $\\Delta $-valued increasing sequence $ \\{\n\t\t\\alpha_{n} \\}_{ n= 1 }^\\infty $ such that\n\t\t$ \\tau(\\omega) = \\lim_{n \\to \\infty} \\alpha_n(\\omega)$, for all $\\omega \\in \\Omega$.\n\\item[(2)]\n\t\tAssume also that for each $ \\alpha \\in \\Delta $, $ I_{\n\t\t\t\\alpha } : [ 0, \\alpha ) \\times \\Omega \\to V $ is an admissible\n\t\tprocess such that for all $ \\alpha , \\beta \\in \\Delta $ and\n\t\tevery $t>0$,\n\t\t\\begin{equation} \\label{eqn-amalgamation_01} I_{ \\alpha }(t)= I_{ \\beta } (t) \\mbox{ $\\mathbb{P}$-a.s. on } \\Omega_t( \\alpha \\wedge \\beta) .\n\t\t\\end{equation}\n\t\tThen, there exists an admissible process $ \\mathbf{I}: [0, \\tau )\n\t\t\\times \\Omega \\to V $,\n\t\tsuch that every $t>0$,\n\t\t\\begin{equation} \\label{eqn-amalgamation_02}\n\t\t\\mathbf{I}(t)= I_{ \\alpha } (t) \\; \\mbox{$\\mathbb{P}$-a.s. on} \\; \\Omega_t( \\alpha ).\n\t\t\\end{equation}\n\\item[(3)]\t\tMoreover,\\hdoubtz{This statement is not in Elworthy's book. Ay least I cannot find it.}\n\t\tif ${\\tilde I} :[0,\\tau) \\times \\Omega \\to X$ is any process\n\t\tsatisfying \\eqref{eqn-amalgamation_02} then the process $\\tilde I$\n\t\tis a version of the process $I$, i.e. for all $t\\in [0,\\infty) $\n\t\t\\begin{equation} \\label{eqn-amalgamation_03}\n\t\t\\mathbb{ P}\\left(\\left\\{\\omega \\in \\Omega: t < \\tau(\\omega) , \\;\n\t\tI(t,\\omega)\\not= {\\tilde I}(t,\\omega) \\right\\} \\right) =0 .\n\t\t\\end{equation}\n\t\tIn particular, if in addition ${\\tilde I}$ is an admissible\n\t\tprocess, then\n\t\t\\begin{equation} \\label{eqn-amalgamation_03'}\n\t\t\\mathbf{I}= {\\tilde I}.\n\t\t\\end{equation}\n\\end{trivlist}\t\t\n\t\n\t\t\n\t\\end{lem}\n\t\\begin{Rem}\\label{rem-amalgamation-equiv}\n\t\tLet us note that because both processes $ \\mathbf{I}: [0, \\tau )\n\t\t\\times \\Omega \\to V $ and $ {I}_\\alpha: [0, \\alpha ) \\times \\Omega\n\t\t\\to V $ are admissible (and hence with almost sure continuous\n\t\ttrajectories), and since $\\alpha \\leq \\tau$, condition\n\t\t\\eqref{eqn-amalgamation_02} is equivalent to the following one:\n\t\t\\begin{equation} \\label{eqn-amalgamation_02'}\n\t\t\\mathbf{I}_{\\vert [0,\\alpha)\\times \\Omega }= I_{ \\alpha } .\n\t\t\\end{equation}\n\t\tSimilarly, condition \\eqref{eqn-amalgamation_01} is equivalent\n\t\tto the following one\n\t\t\\begin{equation} \\label{eqn-amalgamation_01'} {I_{ \\alpha }}_{\\vert [0,\\alpha \\wedge \\beta )\\times \\Omega } = {I_{ \\beta }}_{\\vert [0,\\alpha \\wedge \\beta )\\times \\Omega }.\n\t\t\\end{equation}\n\t\t\t\\end{Rem}\n\\begin{proof}[Proof of Lemma \\ref{lem-amalgamation}] Let $\\Delta$ be the family of accessible stopping\n\t\ttimes with values in $[0, \\infty ]$. This set satisfies the assumptions of Lemma \\cite[Lemma III.6A]{Elw_1982}, where the set $\\Delta$ is denoted by $A$. Indeed,\nby Remark \\ref{rem-predictable stopping time}, the supremum of every finite subset of $\\Delta$ belongs to $\\Delta$. Therefore, there exists an $\\mathcal{F}$-measurable function $\\tau:\\Omega\\to [0,\\infty]$ such that\n\\begin{trivlist}\n\\item[(i)] if $\\sigma \\in \\Delta$, then $\\tau\\geq \\sigma$, $\\mathbb{P}$-a.s.;\n\\item[(ii)] if a random variable $\\eta:\\Omega\\to [0,\\infty]$ satisfies $\\tau\\geq \\sigma$, $\\mathbb{P}$-a.s., for all $\\sigma \\in \\Delta$, then\n$\\eta\\geq \\tau$, $\\mathbb{P}$-a.s..\n\\item[(iii)] there exists a sequence $(\\alpha_n)_{n \\in \\mathbb{N}}$ of elements of $\\Delta$ such that \t for all $\\omega \\in \\Omega$, $\\alpha_n(\\omega) \\leq \\alpha_{n+1}(\\omega) \\leq \\tau(\\omega)$ for all $n\\in \\mathbb{N}$ and\n \t$ \\tau(\\omega) = \\lim_{n\\to \\infty} \\alpha_n(\\omega)= \\sup_{n\\in \\mathbb{N}} \\alpha_n(\\omega)$.\n\\end{trivlist}\nMoreover, $\\tau$ is unique in the sense that if $\\hat \\tau$ satisfies the above conditions (i) and (ii), then $\\hat \\tau \\geq \\tau$, $\\mathbb{P}$-a.s..\nHence, since for every $n \\in \\mathbb{N}$, $\\alpha_n$, is an accessible stopping time, by \\cite[Proposition III.5B]{Elw_1982}, Remark \\ref{rem-predictable stopping time} and \\cite[Proposition 4.11]{Metivier_1982} we infer that $\\tau$ an accessible stopping time. This proves part (1) of Lemma \\ref{lem-amalgamation}.\n\nThe proof of parts (2) and (3) is the same as the proof of \\cite[Lemma III 6 B]{Elw_1982}, so we omit it.\n\\end{proof}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\n\n\t\t\\begin{Def}\\label{Def-maxsol}\n\t\tConsider a family $\\mathcal{ LS}$ of all local solution\n\t\t$(u,\\tau)$ to the problem \\eqref{ABS-SPDE-strong}. For two\n\t\telements $(u,\\tau), (v,\\sigma) \\in \\mathcal{ LS} $ we write that\n\t\t$(u,\\tau)\\preceq (v,\\sigma)$ iff $\\tau \\leq \\sigma$ $\\mathbb{P}$-a.s. and\n\t\t$v_{\\vert [0,\\tau)\\times \\Omega} \\sim u$. Note that if\n\t\t$(u,\\tau)\\preceq (v,\\sigma)$ and $(v,\\sigma)\\preceq (u,\\tau)$,\n\t\tthen $(u,\\tau)\\sim (v,\\sigma)$. We write $(u,\\tau)\\prec\n\t\t(v,\\sigma)$ iff $(u,\\tau)\\preceq (v,\\sigma)$ and $(u,\\tau)\\not\\sim\n\t\t(v,\\sigma)$. Then, the pair $(\\mathcal{ LS},\\preceq)$ is \n\t\tpartially ordered. \n\t\tEach maximal element $(u,\\tau)$ in the set $(\\mathcal{\n\t\t\tLS},\\preceq)$\n\t\tis called a maximal local solution to the problem \\eqref{ABS-SPDE-strong}. The existence of an upper bound of every non-empty chain of $(\\mathcal{\n\t\t\tLS},\\preceq)$ is justified by Amalgamation\n\t\tLemma \\ref{lem-amalgamation}. \\\\\n\t\tIf $(u, \\tau)$ is a maximal local solution to equation\n\t\t\\eqref{ABS-SPDE-strong}, the stopping time $\\tau$ is called its\n\t\tlifetime.\n\t\\end{Def}\n\t\n\t\tA priori, there may be many maximal elements in $(\\mathcal{\n\t\t\tLS},\\preceq)$ and hence many maximal local solutions to\n\t\tthe problem \\eqref{ABS-SPDE-strong}. However, if the uniqueness of local solutions holds, then the\n\t\tuniqueness of the maximal local solution will follow.\n\n\t\\begin{Def}\\label{Def-uniq} A local solution $(u,\\tau)$\n\t\tto problem \\eqref{ABS-SPDE-strong} is unique iff for all other\n\t\tlocal solution $(v,\\sigma)$ to \\eqref{ABS-SPDE-strong} the\n\t\trestricted processes $u_{[0, \\tau\\wedge \\sigma)\\times \\Omega}$ and\n\t\t$v_{[0, \\tau\\wedge\\sigma)\\times \\Omega}$ are equivalent.\n\t\\end{Def}\n\t\n\t\\begin{prop}\\label{prop-loc-implies-max}\n\t\tSuppose that $u_0$ is a $V$-valued random variable and $\\mathcal{F}_0$-measurable. Assume that the following two conditions are satisfies:\n\\begin{trivlist}\n\\item[(i)] there\n\t\texist at least one local solution $(u^0,\\tau^0)$ to problem\n\t\t\\eqref{ABS-SPDE-strong}\n\\item[(ii)]\nif $(u^1,\\tau^2)$ and $(u^2,\\tau^2)$ are local solutions, then for every $t>0$,\n\t\t\\begin{equation} \\label{eqn-amalgamation_04} u^1(t)= u^2 (t) \\mbox{ $\\mathbb{P}$-a.s. on } \\Omega_t( \\tau^1 \\wedge \\tau^2 ).\n\t\n\t\t\\end{equation}\n\\end{trivlist}\nThen, problem\nproblem \\eqref{ABS-SPDE-strong} has a unique maximal local solution { $(\\hat{u},\\hat{\\tau})$} satisfying $(u^0,\\tau^0)\\preceq (\\hat{u},\\hat{\\tau}) $. \n\t\\end{prop}\n\t\n\t\\begin{Rem}\\label{rem--loc-implies-max-equiv}\n\t\tLet us note that similarly to Remark \\ref{rem-amalgamation-equiv},\n\t\tbecause both the local solutions $u^1$ and $u^2$ are admissible\n\t\t(hence with almost sure continuous trajectories), condition\n\t\t\\eqref{eqn-amalgamation_04} is equivalent to \n\t\t\\begin{equation} \\label{eqn-amalgamation_04'}\n\t\tu^1_{\\vert [0,\\tau^1 \\wedge \\tau^2)\\times \\Omega }= u^2_{\\vert\n\t\t\t[0,\\tau^1 \\wedge \\tau^2)\\times \\Omega } .\n\t\t\\end{equation}\n\t\\end{Rem}\n\t\\begin{proof}[Proof of Proposition \\ref{prop-loc-implies-max}]\n\t\tLet us choose and fix a local solution $(u^0,\\tau^0)$ to problem\n\t\t\\eqref{ABS-SPDE-strong} and let us consider the family $\\mathcal{ LS}$\nof all local solution\n\t\t$(u,\\tau)$ to the problem \\eqref{ABS-SPDE-strong} such that $(u^0,\\tau^0)\\preceq (u,\\tau) $.\nBy assumptions this set is non-empty. Due to the assumptions (i) and (ii) of Proposition \\ref{prop-loc-implies-max}, by the Amalgamation Lemma \\ref{lem-amalgamation} we infer that there exists an accessible stopping time\n\t\t\t\\[\n\t\t\\hat{\\tau} := \\sup\\left\\{ \\tau: (u,\\tau)\\in \\mathcal{ LS}\\right\\}\n\t\t\\]\n\t\tand an admissible process $\\hat{u}: [0, \\hat{\\tau} )\n\t\t\\times \\Omega \\to V $,\n\t\tsuch that for all $(u,\\tau)\\in \\mathcal{LS}$ and for $t>0$,\n\t\t\\begin{equation} \\label{eqn-amalgamation_02-a}\n\t\t\\hat{u}(t)= u (t) \\; \\mbox{$\\mathbb{P}$-a.s. on} \\; \\Omega_t( \\tau).\n\t\t\\end{equation}\nMoreover, there exists an increasing sequence $(\\tau_n)$ of accessible stopping times such that $ \\tau(\\omega) = \\lim_{n \\to \\infty} \\tau_n(\\omega)$, for all $\\omega \\in \\Omega$.\n\nIn order to complete the proof of the existence of a maximal lcoal solution, we shall prove that $(\\hat{u},\\hat{\\tau})\\in \\mathcal{LS}$. For this aim, we closely follow the proof of \\cite[Theorem 2.26]{Brz+Elw_2000}.\n Let us define an auxiliary process \t$\\hat{\\eta}=\t\\bigl(\\hat{\\eta}(t)\\bigr),$ $t\\in [0,\\tau)$, such that for each $n\\in \\mathbb{N}$ and $t\\geq 0$, the following equality holds $\\mathbb{P}$-a.s.\n\t\t\\begin{equation}\\label{eqn-proof-02}\n\t\\begin{split}\n\t\\hat{\\eta}(t \\wedge \\tau_n)=S(t\\wedge \\tau_n)u_0 +\\int_0^{t\\wedge \\tau_n} S(t-s) F(\\hat{u}(s\\wedge \\tau_n) )\\, ds +I_{\\tau_n}(t\\wedge \\tau_n),\n\t\\end{split}\n\t\t\\end{equation}\nwhere $I_{\\tau_n}$ is a continuous $V$-valued process process defined by\n\t\t\t\\begin{equation}\\label{eqn-proof-03}\n\t\t\t\\begin{split}\nI_{\\tau_n}(t):= \\int_0^t\\mathds{1}_{[0,\\tau_n)}(s){S}(t-s) G(\\hat{u}(s \\wedge \\tau_m))\\, \t\t\td{W}(s),\\;\\; t\\geq 0.\n\t\t\t\\end{split}\n\t\t\t\\end{equation}\nAssume that $(u,\\tau)\\in \\mathcal{LS}$. \t Define a process $\\eta=\\bigl(\\eta(t)\\bigr)$, $t\\in [0,\\tau)$ by the above formulae \\eqref{eqn-proof-02}-\\eqref{eqn-proof-03} with\n$\\hat{u}$ replaced by $u$ and the announcing sequence $(\\tau_n)$ of the accessible stopping time $\\hat{\\tau}$ replaced by announcing sequence of the accessible stopping time $\\tau$.\nBecause $(u,\\tau)$ is a local solution, we infer that the process\n$\\eta(t)$, $t\\in [0,\\tau)$\n is a version of the process $u(t)$, $t \\in [0,\\tau)$. Since $\\hat{u}$ satisfies \\eqref{eqn-amalgamation_02-a} and assumption (ii) of Proposition \\ref{prop-loc-implies-max} is satisfied,\n {we infer that\n\\begin{equation} \\label{eqn-amalgamation_02-b}\n\t\t\\hat{\\eta}(t)= u (t) \\; \\mbox{$\\mathbb{P}$-a.s. on} \\; \\Omega_t( \\tau).\n\t\t\\end{equation} }\n Hence, by the part (3) of Lemma \\ref{lem-amalgamation}, we infer that the process $\\hat{\\eta}(t)$, $t\\in [0,\\hat{\\tau})$, is a version of the process $\\hat{u}(t)$, $t\\in [0,\\hat{\\tau})$ and therefore we can replace $\\hat{\\eta}$ by $\\hat{u}$ on the LHS of \\eqref{eqn-proof-02}. Therefore, we deduce that $(\\hat{u}, \\hat{\\tau})\\in \\mathcal{LS}$. This completes the existence of a local maximal solution.\n\nAs a byproduct of the above proof of the existence of a local maximal solution $(\\hat{u}, \\hat{\\tau})$ we showed that $(\\hat{u}, \\hat{\\tau})\\in \\mathcal{LS}$. This, in conjunction with the definition of \n\t$\\mathcal{LS}$ implies that $(u^0,\\tau^0)\\preceq (\\hat{u},\\hat{\\tau}) $.\n\nIt remains to prove the uniqueness of the local maximal solutions. For this aim let us suppose that $(u^1,\\tau^1)$ and $(u^2,\\tau^2)$ are two local maximal solutions. Let us put $\\tilde{\\tau} = \\tau^1\\vee \\tau^2$. Then, by part (ii) of Remark \\ref{rem-predictable stopping time} $\\tilde{\\tau}$ is an accessible stopping time with announcing sequence $(\\tilde{\\tau}_n:=\\tau^1_n \\vee \\tau^2_n)_{n \\in \\mathbb{N}}$, where\n$(\\tau^i_n)_{n \\in \\mathbb{N}}$, $i=1,2$ is an announcing sequence of $\\tau^i$. By the uniqueness assumption (ii), we infer that\n\\begin{equation}\\label{eqn-equivalence}\n({u_1}_{\\lvert _{[0,\\tau^1\\wedge \\tau^2) }},\\tau^1\\wedge \\tau^2) \\sim ({u_2}_{\\lvert _{[0,\\tau^1\\wedge \\tau^2) } },\\tau^1\\wedge \\tau^2).\n\\end{equation}\n We shall now prove that $\\tau^1= \\tau^2$ $\\mathbb{P}$-a.s.. Suppose by contradiction that\n\t\t$\\mathbb{P} (\\{\\tau^1\\neq \\tau^2 \\} )>0. $\n\t\tLet $\\Omega_1:= \\{\\tau^1 \\ge \\tau^2 \\}$ and $\\Omega_2:=\\{ \\tau^2 > \\tau^1 \\}$.\n\t\tWe define a process $(\\tilde{u}, \\tilde{\\tau})$ by the following formula\n\t\t\\begin{equation}\\label{eqn-Def-tildeu}\n\t\\tilde{u} (t,\\omega)=\n\t\t\\begin{cases}\n\t\tu^1(t,\\omega) \\text{ if } \\omega \\in \\Omega_1 \\text{ and } t\\in [0, \\tau^1(\\omega) )\\\\\n\t\tu^2(t,\\omega) \\text{ if } \\omega \\in \\Omega_2 \\text{ and } t\\in [0,\\tau^2(\\omega)). \\\\\n\t\t\\end{cases}\n\t\t\\end{equation}\nWe now claim that the process $(\\tilde{u}, \\tilde{\\tau})$ is a local solution to Problem \\eqref{ABS-SPDE-strong}. Let us fix $n \\in \\mathbb{N}$ and $t\\ge 0$. By symmetry, we can assume that $\\tau^1_n(\\omega)\\le \\tau^2_n(\\omega)$ for all $\\omega \\in \\Omega$ and $n\\in \\mathbb{N}$. Firstly, the proof of the admissibility of $\\tilde{u}(t),\\; t \\in [0,\\tilde{\\tau})$ is very similar to the proof in \\cite[Corollary 2.28]{Brz+Elw_2000}.\nSecondly, let us also observe that on $ \\Omega_1$ we have $\\tilde{\\tau}_n< \\tau^1\\wedge \\tau^2$. Hence, we deduce from \\eqref{eqn-Def-tildeu} and \\eqref{eqn-equivalence} that\n\\begin{equation*}\n\\begin{split}\n\\tilde{u}(t\\wedge\\tilde{\\tau}_n)=&\\tilde{u}(t\\wedge \\tau^2_n)=u^1(t\\wedge \\tau^2_n)=u^2(t\\wedge\\tau^2_n)\\\\\n=&S(t\\wedge\\tau^2_n)u_0 +\\int_0^{t\\wedge\\tau^2_n} S(t\\wedge\\tau^2_n-s) F(u^2(s))\\, ds\n+ I^2_{\\tau_n^2}(t \\wedge \\tau^2_n)\\;\\; t\\geq 0,\n\\end{split}\n\\end{equation*}\nwhere\n\\[\n\t\t\tI^2_{\\tau_n^2}(t):= \\int_0^t\\mathds{1}_{[0,\\tau_n^2)}(s){S}(t-s) G(u^2(s \\wedge \\tau_n^2))\\, \t\t\td{W}(s),\\;\\; t\\geq 0.\n\t\t\t\\]\n\nThe last equality follows from the fact that $(u^2,\\tau^2)$ is a local solution. Since $\\tau_n^2=\\tilde{\\tau}_n$, by using \\eqref{eqn-equivalence} and \\cite[Proposition 2.10]{Brz+Elw_2000} we deduce that\n\\begin{equation*}\n\\begin{split}\n\\tilde{u}(t\\wedge\\tilde{\\tau}_n)\n=&S(t\\wedge\\tilde{\\tau}_n)u_0 +\\int_0^{t\\wedge\\tilde{\\tau}_n} S(t\\wedge\\tilde{\\tau}_n-s) F(\\tilde{u}(s))\\, ds\n+ \\tilde{I}_{\\tilde{\\tau_n}}(t\\wedge \\tilde{\\tau_n}), \\;\\; t\\geq 0,\n\\end{split}\n\\end{equation*}\nwhere\n\\[\n\\tilde{I}_{\\tilde{\\tau_n}}(t):=\n\\int_0^t 1_{[0,\\tilde{\\tau}_n)} S(t-s) G(\\tilde{u}(s\\wedge\\tilde{\\tau}_n)) dW(s), \\;\\; t\\geq 0.\n\\]\nHence, $(\\tilde{u},\\tilde{\\tau})$ satisfies equation \\eqref{eq-locsol_00-b} on $\\Omega_1$. In a similar way, we can also show that `$(\\tilde{u},\\tilde{\\tau})$ satisfies \\eqref{eq-locsol_00-b} on $\\Omega_2$. Hence, $(\\hat{u}, \\hat{\\tau})\\in \\mathcal{LS}$.\n\nNow, by construction we have $(u^i, \\tau^i) \\preceq (\\tilde{u}, \\tilde{\\tau})$ for $i\\in \\{1,2\\}$ and there exists $i_0 \\in \\{1,2\\}$ such that\n$ (u^{i_0},\\tau^{i_0}) \\not \\sim (\\tilde{u},\\tilde{\\tau} )$. This contradicts the maximality of $(u^{i_0},\\tau^{i_0})$ and completes the proof of Proposition \\ref{prop-loc-implies-max}.\n\n\t\\end{proof}\nAs a byproduct of the proof of the above Proposition \\ref{prop-loc-implies-max} we deduce the following general result.\n\\begin{cor}\\label{cor-max of two local solutions} Let $(\\mathbf{x},\\sigma)$ and $(\\mathbf{y},\\tau)$\nbe two local\n\t\tsolution to problem \\eqref{ABS-SPDE-strong} such that for every $t>0$,\n\t\t\\begin{equation} \\label{eqn-amalgamation_05} \\mathbf{y}(t)= \\mathbf{x} (t) \\mbox{ $\\mathbb{P}$-a.s. on } \\Omega_t( \\tau \\wedge \\sigma ).\n\t\\end{equation}\nThen the process \t\t $(\\mathbf{z}, \\sigma \\vee \\tau )$ defined by the following formula\n\t\t\\begin{equation}\\label{eqn-Def-tildeu-2}\n\t\\mathbf{z} (t,\\omega)=\n\t\t\\begin{cases}\n\t\t\\mathbf{x}(t,\\omega), \\text{ if } \\sigma(\\omega)\\geq \\tau(\\omega) \\text{ and } t\\in [0, \\sigma(\\omega) ),\\\\\n\t\t\\mathbf{y}(t,\\omega), \\text{ if } \\sigma(\\omega)< \\tau(\\omega) \\text{ and } t\\in [0,\\tau(\\omega)),\n\t\t\\end{cases}\n\t\t\\end{equation}\nis local\n\t\tsolution to problem \\eqref{ABS-SPDE-strong}. The process $(\\mathbf{z}, \\sigma \\vee \\tau )$ is called supremum of $(\\mathbf{x},\\sigma)$ and $(\\mathbf{y},\\tau)$.\n\\end{cor}\n\n\n\n\n\n\t\\subsection{An abstract result}\n\tIn this subsection we prove by a fixed point method some results about the \n\texistence and uniqueness of maximal local mild solution to \\eqref{ABS-SPDE-1}. \n\t\n\tLet\n\t$\\theta:\\mathbb{R}_+\\to [0,1]$ be a ${\\mathcal C}^\\infty_c$ non\n\tincreasing function\n\tsuch that\n\t\\begin{equation}\\label{eqn-theta} \\inf_{x\\in\\mathbb{R}_+}\\theta^\\prime(x)\\geq -1, \\quad \\theta(x)=1\\;\n\t\\mbox{\\rm iff } x\\in [0,1]\\quad \\mbox{\\rm and } \\theta(x)=0 \\;\n\t\\mbox{\\rm iff } x\\in [2,\\infty).\n\t\\end{equation}\n\tand for $n\\geq 1$ set $\\theta_n(\\cdot)=\\theta(\\frac{\\cdot}{n})$.\n\tNote that if $h:\\mathbb{R}_+\\to\\mathbb{R}_+$ is a non decreasing\n\tfunction, then\n\t\\begin{equation}\\label{ineq-theta}\n\t\\theta_n(x)h(x) \\leq h(2n),\\quad\n\n\t\\vert \\theta_n(x)-\\theta_n(y)\\vert \\leq \\frac1n |x-y|, \\text{ for every $x,y\\in {\\mathbb R}$}.\n\t\\end{equation}\n\n\n\n\t\\begin{prop}\\label{prop-global Lipschitz-F}\n\t\tLet $F$ be a mapping satisfying Assumption\n\t\t\\ref{assum-F}. Assume that $\\delta\\in [0,T]$, $\\mathbf{a} \\in X_\\delta$. Then the map\n\t\t\\[\n\t\t\\Phi_{{\\delta,T,\\mathbf{a}}}^n: \\hat{X}_{\\delta,T,\\mathbf{a}} \\ni u \\mapsto \\theta_n( \\vert u\n\t\t\\vert_{X_\\cdot}) F(u) \\in L^2(0,T;H).\n\t\t\\]\n\tis globally Lipschitz and moreover, for all\n\t\t$u_1,u_2 \\in \\hat{X}_{\\delta,T,\\mathbf{a}}$,\n\t\t\\begin{equation}\\label{eqn-global Lipschitz-F}\n\t\t\\vert \\Phi_{\\delta,T,\\mathbf{a}}^n(u_1)-\\Phi_{\\delta,T,\\mathbf{a}}^n(u_2)\\vert_{L^2(0,T;H)} \\leq \t\tC( C +1)\n\\sum_{i=1}^N (2n)^{p_i+2} (T-\\delta)^{(1-\\alpha_i)\/2}\n\\vert u_1 -u_2\n\t\t\\vert_{X_T}.\n\t\t\\end{equation}\n\t\nIn particular, the Lipschitz constant of $\\Phi_{{\\delta,T,\\mathbf{a}}}^n$ is independent of $\\mathbf{a}$.\n\t\\end{prop}\n\n\tThe proof is based on a proof from\n\t\\cite{Brz+Millet_2012} which in turn was based on a proof from \\cite{deBouard+Deb_1999,deBouard+Deb_2003}.\n\tFor simplicity of notation, below we will write $\\Phi_{T}$ instead of $\\Phi_{\\delta,T,\\mathbf{a}}^n$.\n\n\n\t\\begin{proof}[Proof of Proposition \\ref{prop-global Lipschitz-F}] Wlog we can assume that $N=1$ and we will use notation $\\alpha=\\alpha_1$ and $p=p_1$. In this case, the inequality\n\\eqref{eqn-global Lipschitz-F} takes the following form. For every $n\\in \\mathbb{N}$ there exists $C(n)>0$ such that for all $T>\\delta\\geq 0$, all $\\mathbf{a} \\in X_\\delta$ and\nall $u_1,u_2 \\in \\hat{X}_{\\delta,T,\\mathbf{a}}$,\n\t\t\\begin{equation}\\label{eqn-global Lipschitz-F-simple}\n\t\t\\vert \\Phi_{\\delta,T,\\mathbf{a}}^n(u_1)-\\Phi_{\\delta,T,\\mathbf{a}}^n(u_2)\\vert_{L^2(0,T;H)} \\leq \t\tC(n) (T-\\delta)^{(1-\\alpha)\/2}\n\\vert u_1 -u_2\n\t\t\\vert_{X_T}.\n\t\t\\end{equation}\nIn what follows we will prove \\eqref{eqn-global Lipschitz-F-simple}.\nLet us fix $n\\in \\mathbb{N}$, $T>\\delta\\geq 0$, $\\mathbf{a} \\in X_\\delta$ and $u_1,u_2 \\in \\hat{X}_{\\delta,T,\\mathbf{a}}$,\n\nNote that $\\Phi_T(0)=0$. Assume that $u_1,u_2 \\in X_T$. Denote,\n\t\tfor $i=1,2$,\n\t\t\\[\n\t\t\\tau_i= \\inf\\{t \\in [0,T]: \\vert u_i \\vert_{X_t} \\geq 2n\\}.\n\t\t\\]\n\t\tNote that if the set on the RHS above is empty, i.e. $\\vert u_i \\vert_{X_t} < 2n$ for all $t\\in [0,T]$,\n\t\tthen $\\tau_i=T$.\n\t\t\n\t\tWlog we can assume that $\\tau_1 \\leq\n\t\t\\tau_2$. Because for $i=1,2, \\;\\; \\theta_n( \\vert u_i\n\t\t\\vert_{X_t}) =0 \\mbox{ for } t \\geq \\tau_2$, we have \n\t\\begin{eqnarray*}\n\t\n\t\n\t\t\\vert \\Phi_T(u_1)-\\Phi_T(u_2)\\vert_{L^2(0,T;H)} \t&=& \\Big[ \\int_0^{\\tau_2} \\vert \\theta_n( \\vert u_1 \\vert_{X_t})\n\t\t\tF(u_1(t))-\\theta_n( \\vert u_2 \\vert_{X_t})\n\t\t\tF(u_2(t))\\vert_H^2\\,dt\\Big]^{1\/2}\n\t\t\t\\\\\n\t\t\t& &\\hspace{-3truecm}\\lefteqn{ \\leq\n\t\t\t\t\\Big[ \\int_0^{\\tau_2} \\vert \\big[ \\theta_n( \\vert u_1 \\vert_{X_t}) - \\theta_n( \\vert u_2 \\vert_{X_t}) \\big] F(u_2(t))\\vert_H^2 \\,dt \\Big]^{1\/2} }\\\\\n\t\t\t&&\\hspace{-2truecm}\\lefteqn{+ \\Big[ \\int_0^{\\tau_2} \\vert\n\t\t\t\t\\theta_n( \\vert u_1 \\vert_{X_t})\\big[ F(u_1(t))- F(u_2(t)) \\big]\n\t\t\t\t\\vert_H^2\\,dt\\Big]^{1\/2} =:A+B}\n\t\t\\end{eqnarray*}\n\t\tNext, since $\\theta_n$ is Lipschitz with Lipschitz constant $2n$ and ${u_1}_{\\lvert_{[0,\\delta]}}={u_2}_{\\lvert_{[0,\\delta]}}=\\mathbf{a}$\n\t\twe have\n\t\t\\begin{eqnarray*}\n\t\nA^2 &=& \\int_0^{\\delta \\wedge \\tau_2} \\vert \\big[ \\theta_n( \\vert u_1 \\vert_{X_t}) - \\theta_n( \\vert u_2 \\vert_{X_t}) \\big] F(u_2(t))\\vert^2 \\,dt\n+ \\int_{\\delta \\wedge \\tau_2}^{ \\tau_2} \\vert \\big[ \\theta_n( \\vert u_1 \\vert_{X_t}) - \\theta_n( \\vert u_2 \\vert_{X_t}) \\big] F(u_2(t))\\vert^2 \\,dt\n\\\\\n&=& \\int_{\\delta \\wedge \\tau_2}^{ \\tau_2} \\vert \\big[ \\theta_n( \\vert u_1 \\vert_{X_t}) - \\theta_n( \\vert u_2 \\vert_{X_t}) \\big] F(u_2(t))\\vert^2 \\,dt\n\\le 4n^2 C^2 \\int_{\\delta \\wedge \\tau_2}^{ \\tau_2} \\big[ \\vert\n\t\t\t\\vert u_1 \\vert_{X_t} - \\vert u_2 \\vert_{X_t} \\vert\\big]^2 \\vert F(u_2(t))\\vert_H^2 \\,dt\\\\\n\t\n\t\t\t&\\leq& 4n^2 C^2 \\int_{\\delta \\wedge \\tau_2}^{ \\tau_2} \\vert\n\t\t\tu_1 -u_2 \\vert_{X_t}^2 \\vert F(u_2(t))\\vert_H^2 \\,dt \\leq 4n^2 C^2\\vert\n\t\t\tu_1 -u_2 \\vert_{X_T}^2 \\int_{\\delta \\wedge \\tau_2}^{ \\tau_2} \\vert F(u_2(t))\\vert_H^2 \\,dt.\n\t\t\\end{eqnarray*}\n\t\tNext, by assumptions and some elementary calculations \n\t\t\\begin{align*}\n\t\t\\int_{\\delta \\wedge \\tau_2}^{ \\tau_2} \\vert F(u_2(t))\\vert_H^2 \\,dt \\leq & C^2 \\int_{\\delta \\wedge \\tau_2}^{ \\tau_2} \\Vert u_2(t)\\Vert^{2p+2-2\\alpha} \\vert u_2(t)\\vert_E^{2\\alpha} \\,dt\\\\ \n\t\t\\leq & C^2 \\sup_{t \\in [\\delta\\wedge \\tau_2,\\tau_2]} \\Vert u_2(t)\\Vert^{2p+2-2\\alpha}\n\t\t\\big( \\int_{\\delta \\wedge \\tau_2}^{ \\tau_2} \\vert u_2(t)\\vert_E^{2} \\,dt\\big)^\\alpha\n\t\t(\\tau_2-\\delta\\wedge \\tau_2)^{1-\\alpha}\\\\\n\t\t\\leq & C^2 (T-\\delta)^{1-\\alpha} \\vert u_2 \\vert_{X_{\\delta\\wedge\\tau_2, \\tau_2}}^{2p+2} \\leq C^2 (T-\\delta)^{1-\\alpha}(2n)^{2p+2}.\n\t\t\\end{align*}\n\t\tTherefore,\n\t\t\\begin{equation*}\n\t\t\tA\\le C (T-\\delta)^{(1-\\alpha)\/2}(2n)^{p+2} \\vert u_1 -u_2\n\t\t\t\\vert_{X_T} .\n\t\t\\end{equation*}\n\tSince $\\tau_1 \\leq \\tau_2$, $ \\theta_n( \\vert u_1 \\vert_{X_t}) =0$, $ t \\geq\n\t\t\\tau_1$, $\\theta_n( \\vert u_1 \\vert_{X_t}) \\leq 1,\\, t\\in [0,\\tau_1)$ and ${u_1}_{\\lvert_{[0,\\delta]}}={u_2}_{\\lvert_{[0,\\delta]}}=\\mathbf{a}$, we have\n\t\t\\begin{align*}\n\t\t\tB= \\Big[ \\int_0^{\\tau_2} \\vert \\theta_n( \\vert u_1 \\vert_{X_t})\\big[ F(u_1(t))- F(u_2(t)) \\big] \\vert_H^2\\,dt \\Big]^{1\/2}\n\t\n\t\t\t\\leq& \\Big[\\int_{\\delta\\wedge \\tau_1}^{\\tau_1} \\vert F(u_1(t))- F(u_2(t)) \\vert_H^2\\,dt\\Big]^{1\/2}\n\t\t\t\\leq \\tilde{B}_{p},\n\t\t\\end{align*}\n\t\twhere\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\tilde{B}_{p}:=\n\t\tC \\Big[ \\int_{\\delta\\wedge \\tau_1}^{\\tau_1} \\vert u_1(t)-u_2(t)\\vert_E^{2\\alpha} \\Vert u_1(t)-u_2(t)\\Vert^{2-2\\alpha} \\Vert u_2(t)\\Vert^{2p} \\,dt\\Big]^{1\/2}\\\\\n\t\t+C \\Big[ \\int_{\\delta\\wedge \\tau_1}^{\\tau_1} \\Vert u_1(t)-u_2(t)\\Vert^2 \\Vert u_1(t)\\Vert^{2p-2\\alpha} \\vert u_1(t)\\vert_E^{2\\alpha} \\,dt\\Big]^{1\/2}.\n\t\t\\end{split}\n\t\t\\end{equation*}\nThis term can be estimated as follows\n\t\t\\begin{align*}\n\t\t\\tilde{B}_{p} \t\\leq& C \\sup_{t\\in [0,T]} \\Vert u_1(t)-u_2(t)\\Vert \\sup_{t\\in\n\t\t\t\t[\\delta\\wedge \\tau_1,\\tau_1]} \\Vert u_1(t)\\Vert^{p-\\alpha} \\Big[ \\int_{\\delta\\wedge \\tau_1}^{\\tau_1}\n\t\t\t\\vert u_1(t)\\vert_E^{2} \\,dt\\Big]^{\\alpha\/2}\n\t\t\t(\\tau_1-\\delta\\wedge \\tau_1)^{(1-\\alpha)\/2}\n\t\t\t\\\\\n\t\t\t+& C \\sup_{t\\in [0,T]} \\Vert u_1(t)-u_2(t)\\Vert^{1-\\alpha}\n\t\t\t\\sup_{t\\in [\\delta\\wedge \\tau_1,\\tau_1]} \\Vert u_2(t)\\Vert^p \\Big[ \\int_{\\delta\\wedge \\tau_1}^{\\tau_1}\n\t\t\t\\vert u_1(t)-u_2(t)\\vert_E^{2} \\,dt\\Big]^{\\alpha\/2}\n\t\t\t(\\tau_1-\\delta\\wedge \\tau_1)^{(1-\\alpha)\/2}\n\t\t\t\\\\\n\t\t\t\\leq& C \\vert u_1-u_2\\vert_{X_T} \\vert u_1 \\vert_{X_{\\tau_1}}^p (T-\\delta)^{(1-\\alpha)\/2} +C \\vert u_1-u_2\\vert_{X_T} \\vert u_2 \\vert_{X_{\\tau_1}}^p (T-\\delta)^{(1-\\alpha)\/2}\\\\\n\t\n\t\t\t\\leq& C (T-\\delta)^{(1-\\alpha)\/2} \\vert u_1-u_2\\vert_{X_T} \\Big[\n\t\t\t\\vert u_1 \\vert_{X_{\\tau_1}}^p + \\vert u_2\n\t\t\t\\vert_{X_{\\tau_1}}^p\\Big] \\leq C (2n)^{p+1} (T-\\delta)^{(1-\\alpha)\/2}\n\t\t\t\\vert u_1-u_2\\vert_{X_T}.\n\t\t\\end{align*}\n\t\tSumming up, we proved the following inequality \n\t\t\\begin{eqnarray*}\n\t\t\t\\vert \\Phi_T(u_1)-\\Phi_T(u_2)\\vert_{L^2(0,T;H)}\\le C\\Big[ 2n C +1\\Big] (2n)^{p+1} \\tau_2^{(1-\\alpha)\/2} \\vert\n\t\t\tu_1 -u_2 \\vert_{X_T},\n\t\t\\end{eqnarray*}\n\t\twhich competes the proof of the proposition.\n\t\\end{proof}\n\tThe following result is a special case of Proposition\n\t\\ref{prop-global Lipschitz-F} with $H=V$.\n\t\\begin{cor}\\label{cor-global-lip-G}\n\t\tLet $G$ be a nonlinear mapping satisfying Assumption\n\t\t\\ref{assum-G}. Assume that $n\\in \\mathbb{N}$, $T>0$, $\\delta \\in [0,T]$ and $\\mathbf{a}\\in X_{0,\\delta}$.\n Define a map $\\hat{\\Phi}_{{\\delta,T,\\mathbf{a}}}^n$ by\n\t\t\\begin{equation}\\label{eqn-Phi_G}\n\t\t\\hat{\\Phi}_{{\\delta,T,\\mathbf{a}}}^n: \\hat{X}_{\\delta,T,\\mathbf{a}} \\ni u \\mapsto \\theta_n(\n\t\t\\vert u \\vert_{X_\\cdot}) G(u) \\in L^2(0,T;V).\n\t\t\\end{equation}\n\t\tThen $\\hat{\\Phi}_{{\\delta,T,\\mathbf{a}}}^n$ is globally Lipschitz and moreover, for all\n\t\t$u_1,u_2 \\in \\hat{X}_{\\delta,T,\\mathbf{a}}$,\n\t\t\\begin{eqnarray}\\label{eqn-global LipschitzG}\n\t\t\\vert \\hat{\\Phi}_{{\\delta,T,\\mathbf{a}}}^n(u_1)-\\hat{\\Phi}_{{\\delta,T,\\mathbf{a}}}^n(u_2)\\vert_{L^2(0,T;V)} &\\leq\n\t\t& (2n)^{k+2}C_G(C_G +1) {(T-\\delta)^{(1-\\beta)\/2} }\\vert u_1\n\t\t-u_2 \\vert_{X_T}.\n\t\t\\end{eqnarray}\n{In particular, the Lipschitz constant of $\\hat{\\Phi}_{{\\delta,T,\\mathbf{a}}}^n$ is independent of $\\mathbf{a}$.}\n\t\\end{cor}\n\n\n\n\t\\begin{prop}\\label{prop-Psi_T-Lipschitz}\n\tAssume that Assumptions \\ref{assum-F} and \\ref{assum-01} hold. Assume that $n\\in \\mathbb{N}$, $T>0$, $\\delta \\in [0,T]$ and $\\mathbf{a}\\in \\mathscr{M}^2(X_{0,\\delta})$.\n\tThen the map $\\Psi^n_{\\delta,T,\\mathbf{a}}$ defined by\n\t\t\\begin{equation}\\label{eqn-Psi_T}\n\\Psi_{\\delta,T,\\mathbf{a}}^n: \\mathscr{M}^2(\\hat{X}_{\\delta,T, \\mathbf{a}}) \\ni u \\mapsto [S(\\cdot)](\\mathbf{a}(0)) + S \\ast \\Phi_{\\delta, T,\\mathbf{a}}^n\n\t\t(u)+ S\\diamond \\hat{\\Phi}_{\\delta,T,\\mathbf{a}}^n(u) \\in \\mathscr{M}^2(X_{T}),\n\t\t\\end{equation}\n\t\t is globally Lipschitz and moreover, for all $u_1,u_2\n\t\t\\in \\mathscr{M}^2(\\hat{X}_{\\delta,T, \\mathbf{a}})$,\n\t\t\\begin{eqnarray}\\label{eqn-global Lipschitz}\n\t\t\\vert \\Psi_{\\delta,T,\\mathbf{a}}^{n}(u_1)-\\Psi_{\\delta,T,\\mathbf{a}}^{n}(u_2)\\vert_{\\mathscr{M}^2(X_T)} \\leq \\hat{C}(n) \\Big[\\max_{1\\le i \\le N}{(T-\\delta)^{1-\\alpha_i}} \\vee (T-\\delta)^{1-\\beta} \\Big]^{\\frac12}\\vert u_1 -u_2\n\t\t\\vert_{\\mathscr{M}^2(X_T)},\n\\nonumber\n\t\t\\end{eqnarray}\n\t\twhere $\\hat{C}(n)$ is dependent only on $n$ and is given by, for some $D>1$,\n\t\t\\[ \\hat{C}(n)= C_1 C_F (C_F +1) \\sum_{i=1}^N(2n)^{p_i+2} + C_2C_G (2n)^{k+2}( C_G +1)\n\\leq C_3n^{D}.\n\t\t\\]\nIn particular, the Lipschitz constant of $\\Psi_{\\delta,T,\\mathbf{a}}^{n}$ is independent of $\\mathbf{a}$.\n\t\\end{prop}\n\t\\begin{proof}[Proof of Proposition \\ref{prop-Psi_T-Lipschitz}]\n\nFor simplicity of notation we will write\n\t\t$\\Psi_{T}$ instead of $\\Psi_{\\delta,T,\\mathbf{a}}^n$. We will also write $\\Phi_F$ (resp. $\\hat{\\Phi}_G$) instead of $\\Phi^n_{\\delta,T,\\mathbf{a}}$ (resp. $\\hat{\\Phi}^n_{\\delta,T,\\mathbf{a}}$).\n\t\tObviously in view of Assumption \\ref{assum-01} the map $\\Psi_T$ is\n\t\twell defined. Let us fix $u_1,u_2 \\in \\mathscr{M}^2(X_{\\delta,T,\\mathbf{a}})$. Then by the Fubini Theorem, Assumption \\ref{assum-01}, Proposition \\ref{prop-global Lipschitz-F} and Corollary \\ref{cor-global-lip-G} we infer that\n\t\t\\begin{eqnarray*}\n\t\t\t\\vert \\Psi_T(u_1)-\\Psi_T(u_2)\\vert_{\\mathscr{M}^2(X_T)} &\\leq & \\vert S\\ast \\Phi_F(u_1)-S\\ast \\Phi_F(u_2)\\vert_{\\mathscr{M}^2(X_T)}\n\t\t\t+ \\vert S \\diamond \\Phi_G(u_1)-S\\diamond \\Phi_G(u_2)\\vert_{\\mathscr{M}^2(X_T)}\\\\\n\t\t\t&\\leq& C_1 \\vert \\Phi_F(u_1)- \\Phi_F(u_2)\\vert_{\\mathscr{M}^2(0,T;H)}+C_2\n\t\t\t\\vert \\Phi_G(u_1)-\\Phi_G(u_2)\\vert_{\\mathscr{M}^2(X_T)}\n\t\t\t\\\\\n\t\t\t&\\leq& \\hat{C}(n)\\Big[\\max_{1\\le i \\le N}{(T-\\delta)^{1-\\alpha_i}} \\vee (T-\\delta)^{1-\\beta} \\Big]^{\\frac12}\\vert u_1 -u_2\n\t\t\t\\vert_{\\mathscr{M}^2(X_T)}.\n\t\t\\end{eqnarray*}\n\t\tThe proof is complete.\n\t\\end{proof}\nSince our method is based on finding fixed points of $\\Psi_{\\delta,T,\\mathbf{a}}^n$, the following auxiliary result is useful.\n\\begin{lem}\\label{lem-fixed points}\nAssume that $n\\in \\mathbb{N}$ and $T>S> 0$ and $\\mathbf{x} \\in V$. Assume that $\\mathbf{a}\\in \\mathscr{M}^2(\\hat{X}_{0,S,\\mathbf{x}})$ is a fixed point of\n$\\Psi_{0,S,\\mathbf{x}}^n$. Then $\\Psi_{S,T,\\mathbf{a}}^n$ maps $\\mathscr{M}^2(\\hat{X}_{S,T, \\mathbf{a}})$ into itself.\n\\end{lem}\n\\begin{proof}[Proof of Lemma \\ref{lem-fixed points}] Let us choose and fix $n\\in \\mathbb{N}$, $T>S> 0$, $\\mathbf{x} \\in V$ and $\\mathbf{a}\\in \\mathscr{M}^2(\\hat{X}_{0,S,\\mathbf{x}})$, a fixed point of\n$\\Psi_{0,S,\\mathbf{x}}^n$.\nWe will show that $\\Psi^n_{S,T,\\mathbf{a} }$ maps $\\mathscr{M}^2(\\hat{X}_{S,T,\\mathbf{a}})$ into itself.\nTake an arbitrary $u\\in \\hat{X}_{S,T,\\mathbf{a}}$. Since by Proposition \\ref{prop-Psi_T-Lipschitz}, $v:=\\bigl[\\Psi_{S,T,\\mathbf{a}}^n \\bigr](u) \\in \\mathscr{M}^2(X_T)$ we only need to show that\n$v_{|[0,S]}=\\mathbf{a}$. For this aim let us observe that by Definition \\ref{eqn-Psi_T}, we have for $t\\in [0,S]$,\n\\begin{align*}\n v(t)= S(t)(\\mathbf{a}(0)) &+ \\int_0^t S(t-r) \\theta_n(\\lvert u\\rvert_{X_r}) F(u(r))\\, dr\n + \\int_0^t S(t-r) \\theta_n( \\lvert u \\rvert_{X_r}) G(u(r))\\, dW(r).\n \\end{align*}\nBecause $u_{|[0,S]}=\\mathbf{a}$, $\\mathbf{a}(0)=\\mathbf{x}$ and, by assumptions $\\Psi_{0,S,\\mathbf{x}}^n(\\mathbf{a})=\\mathbf{a}$ in $\\mathscr{M}^2(\\hat{X}_{0,S, \\mathbf{x}})$, we infer that\n\\begin{align*}\n v(t)= S(t)\\mathbf{x} &+ \\int_0^t S(t-r) \\theta_n(\\lvert a\\rvert_{X_r}) F(a(r))\\, dr\n + \\int_0^t S(t-r) \\theta_n( \\lvert a \\rvert_{X_r}) G(a(r))\\, dW(r)\\\\\n &=[\\Psi_{0,S,\\mathbf{x}}^n(\\mathbf{a})](t)=\\mathbf{a}(t), \\;\\;\\; t\\in [0,S].\n \\end{align*}\nThis completes the proof of Lemma \\ref{lem-fixed points}.\n \\end{proof}\n\n\n\n\\begin{comment}\nBefore proceeding further, we state the following remark.\n\\begin{Rem}\\label{Rem-Global-Lipshitz}\n\t\\begin{trivlist}\n\t\t\\item[(i)] Let $\\mathbf{a}_0\\in V$. Then, the results in Proposition \\ref{prop-global Lipschitz-F} and Corollary \\ref{cor-global-lip-G} remain true with the space $\\hat{X}_{\\delta,T,\\mathbf{a}}$ replaced by\n\t\t$\\hat{X}_{0,T,\\mathbf{a}_0}$.\n\t\t\\item[(ii)] Let $\\mathbf{a}_0\\in L^2(\\Omega,\\mathcal{F}_0;V)$ and $n \\in \\mathbb{N}$. Then, it follows from item (i) and the proof of Proposition \\ref{prop-Psi_T-Lipschitz} that the map $\\Psi_{0,T,\\mathbf{a}_0}^n$ defined by\n\t\t\\begin{equation}\\label{eqn-Psi_T-0}\n\t\t\\Psi_{0,T,\\mathbf{a}_0}:=\\Psi_{0,T,\\mathbf{a}_0}^n: \\mathscr{M}^2(\\hat{X}_{0,T, \\mathbf{a}_0}) \\ni u \\mapsto S\\mathbf{a}_0 + S \\ast \\Phi_{0, T,\\mathbf{a}_0}^n\n\t\t(u)+ S\\diamond \\hat{\\Phi}_{0,T,\\mathbf{a}_0}^n(u) \\in \\mathscr{M}^2(X_{T}),\n\t\t\\end{equation}\n\t\tis globally Lipschitz with the Lipschitz constant being independent of $\\mathbf{a}_0$.\n\t\\end{trivlist}\n\tIn both items (i) and (ii), inequalities \\eqref{eqn-global Lipschitz-F}, \\eqref{eqn-global LipschitzG} and \\eqref{eqn-global Lipschitz} hold with $T-\\delta$ replaced by $T$.\n\t\n\t\\end{Rem}\n\\end{comment}\n\n\n\n\n\tThe \tfirst two main results of this subsection are given in the following two\n\ttheorems.\n\t\\begin{thm}\\label{Thm:LocalUniqueness}\n\t\tSuppose that Assumption \\ref{assum-F}-Assumption \\ref{assum-01} hold. Let $u_0$ be a\t$\\mathcal{F}_{0}$-measurable $V$-valued square integrable random\n\t\tvariable $u_0$ and $(u,\\tau) $ and $(v,\\sigma)$ two local solutions of \\eqref{ABS-SPDE-1}. Then,\n\t\\begin{equation}\n\t(u_{\\lvert_{[0,\\sigma \\wedge \\tau)\\times \\Omega } }, \\sigma \\wedge \\tau)\\sim (v_{\\lvert_{[0,\\sigma \\wedge \\tau)\\times \\Omega}},\\sigma \\wedge \\tau).\n\t\\end{equation}\n\t\\end{thm}\n\\begin{proof}[Proof of Theorem \\ref{Thm:LocalUniqueness}]\n\tWlog we can assume in Assumptions \\ref{assum-F} and \\ref{assum-G} that $N=1$ and $p_i=k$ and will use the notations $\\alpha_1=\\alpha$, $p_1=p=k$.\n\tLet $(u,\\tau) $ and $(v,\\sigma)$ be two local solutions of \\eqref{ABS-SPDE-1}. Let $(\\tau_n)_{n \\in \\mathbb{N}}$ and $(\\sigma_{n \\in \\mathbb{N}})$ be the announcing sequences of $\\tau$ and $\\sigma$, respectively. By \\cite[Propositions 4.3 \\& 4.11 and Theorem 6.6]{Metivier_1982} the stopping time $\\varrho:=\\tau \\wedge \\sigma$ is accessible and it is easy to show that $(\\varrho_n)_{n \\in \\mathbb{N}}:= (\\tau_n \\wedge \\sigma_n)_{n \\in \\mathbb{N}}$ is an announcing sequence of $\\varrho$.\n\t\n\tHereafter we fix $n\\in \\mathbb{N}$. Since $(v,\\sigma)$ is a local solution to \\eqref{ABS-SPDE-strong} and $\\varrho_n \\le \\sigma_n$, by Corollary \\ref{cor-A.2} we infer that for all $t\\ge 0$ $\\mathbb{P}$-a.s.\n\t\\begin{align}\n\tv(t\\wedge \\varrho_n)= & S_{t\\wedge \\varrho_n} u_0 + \\int_0^{t\\wedge \\varrho_n} S_{t\\wedge \\varrho_n-r}F(v(r)) dr + I_{\\sigma_n}(t\\wedge \\varrho_n) \\nonumber \\\\\n\t=& S_{t\\wedge \\varrho_n} u_0 + \\int_0^{t\\wedge \\varrho_n} S_{t\\wedge \\varrho_n-r}F(v(r)) dr + I_{\\varrho_n}(t\\wedge \\varrho_n),\\label{Eq:vstopped is a local sol}\n\t\\end{align}\n\twhere $$ I_{\\sigma_n}(t):=\\int_0^t 1_{[0,\\sigma_n] }(r) S_{t-r}G(v(r))dW(r),\\; t\\ge 0 .$$\n\tThe identity \\eqref{Eq:vstopped is a local sol} proves that $(v,\\sigma\\wedge \\tau)$ is a local solution to \\eqref{ABS-SPDE-strong}. In a similar way, we prove that $(u, \\sigma\\wedge \\tau)$ is a local solution to \\eqref{ABS-SPDE-strong} as well.\n\t\n\tThirdly, for $k \\in \\mathbb{N}$ we put $\\varrho_{n,k}=\\tau_n \\wedge \\sigma_n\\wedge \\tilde{\\tau}_k \\wedge \\tilde{\\sigma}_k$, where\n\t\\begin{align}\n\\tilde{\\tau}_k=\\inf\\{t \\in [0,T]: \\lvert v \\rvert_{X_t} \\ge k \\} \\wedge \\tau \\text{ and } \t\\tilde{\\sigma}_k=\\inf\\{t \\in [0,T]: \\lvert v \\rvert_{X_t} \\ge k \\} \\wedge \\sigma, \\; k \\in \\mathbb{N}.\\nonumber\n\t\\end{align}\nWe observe that for all $k\\in \\mathbb{N}$ $\\varrho_{n,k}\\le \\varrho_{n, k+1}\\le \\sigma_n \\wedge \\tau_n$ $\\mathbb{P}$-a.s. and $\\varrho_{n,k} \\toup \\sigma_n \\wedge \\tau_n$ $\\mathbb{P}$-a.s. if $k \\to \\infty$. Let us now fix $k \\in \\mathbb{N}$. Arguing as in the proof of \\eqref{Eq:vstopped is a local sol} we show that for all $t\\ge 0$, $\\mathbb{P}$-a.s.\n\t\\begin{align}\n\tv(t\\wedge \\varrho_{n,k})\n\t=&S_{t\\wedge \\varrho_{n,k}} u_0 + \\int_0^{t\\wedge \\varrho_{n,k}} S_{t\\wedge \\varrho_{n,k}-r}F(v(r)) dr + I_{\\varrho_{n,k}}(t\\wedge \\varrho_{n,k}).\n\t\\end{align}\n\tIn a similar way, we prove that the same identity holds with $v$ replaced by $u$. Hence, setting $w=u-v$ we infer that for all $t\\ge 0$, $\\mathbb{P}$-a.s.\n\t\\begin{equation}\n\tw(t\\wedge \\varrho_{n,k})\n\t= \\int_0^{t\\wedge \\varrho_{n,k}} S_{t\\wedge \\varrho_{n,k}-r}[F(u(r))- F(v(r))] dr + \\tilde{I}_{\\varrho_{n,k}}(t\\wedge \\varrho_{n,k}).\n\t\\end{equation}\n\twhere\n\t$$ \\tilde{I}_{\\varrho_{n,k}}(t):=\\int_0^t 1_{[0,\\varrho_{n,k}] }(r) S_{t-r}[G(u(r))-G(v(r))]dW(r),\\; t\\ge 0 .$$\n\t\n\tHereafter, $c>0$ denotes an universal constant (independent of $n$ and $k$) which may change from one term to the other.\nFollowing the lines of \\cite[Proof of Lemma 3.8]{Brz+Gat_99} or \\cite[Page 134]{Brz+Masl+Seidler_2005} and using Assumptions \\ref{assum-01}, \\ref{assum-F} and \\ref{assum-G} we infer that for all $t\\ge 0$\n\\begin{align}\n\\mathbb{E} \\lvert w \\rvert^2_{X_{t\\wedge \\varrho_{n,k}} }\\le & c \\mathbb{E} \\int_0^{t\\wedge \\varrho_{n,k}} \\Big[ \\lvert F(u(r)) -F(v(r))\\rvert^2_H + \\lVert G(u(r)) -G(v(r))\\rVert^2\\Big]dr\\nonumber\\\\\n\t\\le & c \\mathbb{E} \\int_0^{t\\wedge \\varrho_{n,k}} \\left(\\Vert w(s) \\Vert^2 \\Vert u(s) \\Vert^{2(p-\\alpha)} \\lvert u(s) \\rvert^{2\\alpha}_E \\right) ds+\n\tc \\mathbb{E}\\int_0^{t\\wedge \\varrho_{n,k}} \\left(\\vert w(s) \\vert^{2\\alpha}_E \\Vert w(s) \\Vert^{2(1-\\alpha)} \\lVert v(s) \\rVert^{2p} \\right)ds\\nonumber \\\\\n\t&=:c\\mathbb{E}I_1+c\\mathbb{E} I_2.\\label{Eq:Uniq-Step-0}\n\\end{align}\nThe H\\\"older inequality and the definition of the stopping time $\\varrho_{n,k}$ imply that for all $t\\ge 0$, $\\mathbb{P}$-a.s.\n\\begin{align}\n I_1\\le & c \\left(\\int_0^{t\\wedge \\varrho_{n,k}} \\Vert w(s)\\Vert^{\\frac{2}{1-\\alpha}} \\Vert u(s)\\Vert^{\\frac{2(p-\\alpha)}{1-\\alpha}} ds \\right)^{1-\\alpha} \\left(\\int_0^{t\\wedge \\varrho_{n,k}} \\vert u(s) \\vert^{2}_E ds \\right)^{\\alpha} \\\\\n\\le & c R^{2\\alpha}\\sup_{s\\in [0,t\\wedge \\varrho_{n,k}]} \\Vert u(s) \\Vert^{2(p-\\alpha)}\\left(\\int_0^{t\\wedge \\varrho_{n,k}} \\Vert w(s)\\Vert^{\\frac{2}{1-\\alpha}} ds \\right)^{1-\\alpha}\\\\\n\\le &c R^{2p} \\lvert w \\rvert_{X_{t\\wedge \\varrho_{n,k}}}^{2\\alpha} \\left(\\int_0^{t\\wedge \\varrho_{n,k}} \\Vert w(s)\\Vert^2 ds \\right)^{1-\\alpha}\n\\le \\frac14 \\lvert w \\rvert_{X_{t\\wedge \\varrho_{n,k}}}^{2} + c R^{\\frac{2p}{1-\\alpha}} \\int_0^{t\\wedge \\varrho_{n,k}} \\Vert w(s)\\Vert^2 ds .\\label{Eq:Uniq-I_1}\n\\end{align}\nIn a similar way one can prove that for all $t\\ge 0$, $\\mathbb{P}$-a.s.\n\\begin{align}\n I_2\\le & c \\left(\\int_0^{t\\wedge \\varrho_{n,k}} \\Vert w(s)\\Vert^2 \\Vert v(s)\\Vert^{\\frac{2p}{1-\\alpha}} ds \\right)^{1-\\alpha} \\left(\\int_0^{t\\wedge \\varrho_{n,k}} \\vert w(s) \\vert^{2}_E ds \\right)^{\\alpha} \\\\\n\\le & \\frac14 \\lvert w \\rvert_{X_{t\\wedge \\varrho_{n,k}}}^{2} + c R^{\\frac{2p}{1-\\alpha}} \\int_0^{t\\wedge \\varrho_{n,k}} \\Vert w(s)\\Vert^2 ds.\\label{Eq:Uniq-I_2}\n\\end{align}\nHence, plugging \\eqref{Eq:Uniq-I_1} and \\eqref{Eq:Uniq-I_2} in \\eqref{Eq:Uniq-Step-0} and using $ \\Vert w(t\\wedge \\varrho_{n,k} )\\Vert^2\\le \\lvert w\\rvert^2_{X_{t\\wedge \\varrho_{n,k}}},\\;$ we infer that\n\\begin{equation}\n\\mathbb{E} \\lVert w(t\\wedge \\varrho_{n,k} )\\rVert^2 \\le 2 c R^{\\frac{2p}{1-\\alpha}} \\int_0^{t\\wedge \\varrho_{n,k}} \\Vert w(s\\wedge \\varrho_{n,k} ) \\Vert^2 ds, \\;\\forall t\\ge0.\n\\end{equation}\nThis along the Gronwall lemma implies that for all $t\\ge 0$, $\n\\mathbb{E} \\Vert w(t\\wedge \\varrho_{n,k}) \\Vert^2=0.$\nHence, by letting $k \\to \\infty$ we infer that for all $t\\ge 0$, $\n\\mathbb{E} \\Vert w(t\\wedge \\tau_n\\wedge \\sigma) \\Vert^2=0,$\n which along with the continuity of $w$ completes the proof of the theorem.\n\\end{proof}\n\n\n\t\\begin{thm}\\label{thm_local} Suppose that Assumptions \\ref{assum-F}-\\ref{assum-01} are satisfied. Then\n\\begin{trivlist}\n\\item[(I)]\n for every\n\t\t$\\mathcal{F}_{0}$-measurable $V$-valued square integrable random\n\t\tvariable $u_0$ there exits a local process $u=\\big(u(t),\n\t\tt\\in[0,T_1) \\big) $ which is the\n\t\tunique local solution to problem \\eqref{ABS-SPDE-strong},\n\\begin{comment}\\item[(II)]\n for all $R>0$ and $\\varepsilon >0$ there exists\n\t\t$\\tau(\\varepsilon,R)>0$, such that for every\n\t\t$\\mathcal{F}_0$-measurable $V$-valued random variable $u_0$\n\t\tsatisfying $\\mathbb{E}\\Vert u_0 \\Vert^{2} \\leq R^{2}$, one has\n\t\t\\begin{equation}\\label{ineq-positive probability}\n{\\mathbb P}\\big(T_1\\geq \\tau(\\varepsilon,R)\\big) \\geq\n\t\t1-\\varepsilon.\\end{equation}\n\\end{comment}\n\\item[(II)]\n if $R>0$ and $\\varepsilon >0$ then there exists a number \n\t\t$T^\\ast(\\varepsilon,R)>0$, such that for every set $\\Omega_1 \\in \\mathcal{F}_0$ and every\n\t\t$\\mathcal{F}_0$-measurable $V$-valued random variable $u_0$ such that\n\t\t \\[\\Vert u_0 \\Vert \\leq R \\mbox{ $\\mathbb{P}$-a.s. on } \\Omega_1,\\] one has\n\t\t\\begin{equation}\\label{ineq-positive probability-2}\n{\\mathbb P}\\big(\\{T_1\\geq T^\\ast(\\varepsilon,R)\\} \\cap \\Omega_1 \\big) \\geq\n\t\t(1-\\varepsilon)\\mathbb{P}(\\Omega_1).\\end{equation}\n\n\n\\end{trivlist}\n\t\\end{thm}\n\t\\begin{proof}[Proof of Theorem \\ref{thm_local}]\n\n\t\tWlog we can assume that $N=1$ and we will use notation $\\alpha=\\alpha_1$ and $p=p_1$.\nLet $u_0\\in\n\t\tL^2(\\Omega, \\mathbb{P}; V)$. We also fix a natural number $n\\in \\mathbb{N}$ in Steps \\textbf{1}-\\textbf{5}.\nIn the first part consisting of Steps \\textbf{1}-\\textbf{7} we will prove the part (I) of the Theorem, i.e. the existence and uniqueness of a local solution to problem \\eqref{ABS-SPDE-strong}.\nPart (II) of the Theorem will be proven in Steps \\textbf{8}-\\textbf{9}.\n\n\\noindent \\textit{Proof of part (I)}\n\\begin{trivlist}\n\\item[\\textbf{Step 1.}] Let us fix $n\\in \\mathbb{N}$ and $T>0$.\nLet $\\Psi^n_{0,T,u_0}: \\mathscr{M}^2(\\hat{X}_{0,T,u_0}) \\to \\mathscr{M}^2(\\hat{X}_{0,T,u_0}) $.\nBy Proposition \\ref{prop-Psi_T-Lipschitz} the map $\\Psi^n_{0, T,u_0}$ is well defined\n and for sufficiently small $T=\\delta_n$, and all $\\mathbf{a}_0$,\n it is an $\\frac12$-contraction. \n\t\t\tThus, by the Banach Fixed Point Theorem, there exists a unique $u^{[n,1]}\\in \\mathscr{M}^2(\\hat{X}_{0,\\delta_n,u_0})$ such that\n\t\t\t\\[u^{[n,1]}=\\Psi^n_{0,\\delta_n,u_0}(u^{[n,1]}).\\] We fix $u^{[n,1]}$ for the rest of the proof. We also put $M:=\\frac{T}{\\delta_n}\\in \\mathbb{N}$.\t\t\n\n\\item[\\textbf{Step 2.}]\nBy Lemma \\ref{lem-fixed points} $\\Psi^n_{\\delta_n, 2 \\delta_n, u^{[n,1]} }$ maps $\\mathscr{M}^2(\\hat{X}_{\\delta_n, 2 \\delta_n, u^{[n,1]}})$ into itself\nand by Proposition \\ref{prop-Psi_T-Lipschitz} and inequality \\eqref{eqn-global Lipschitz} it is an $\\frac 12$-contraction. Therefore, we can find a unique\n $u^{[n,2]} \\in \\mathscr{M}^2(\\hat{X}_{\\delta_n, 2\\delta_n, u^{[n,1]}})$, which we fix for the rest of the proof, such that \\[u^{[n,2]}=\\Psi^n_{\\delta_n, 2\\delta_n, u^{[n,1]}}(u^{[n,2]})\\in \\mathscr{M}^2(X_{\\delta, 2\\delta_n, u^{[n,1]}}).\\]\n\n\n\\item[\\textbf{Step 3.}]\n\t\t\t By induction we can construct a sequence $(u^{[n,k]})_{k=1}^\\infty$ such that\n\\[u^{[n,k]}=\\Psi^n_{(k-1)\\delta_n,k\\delta_n, u^{[n,k-1]}}(u^{[n,k]})\\in \\mathscr{M}^2(X_{(k-1)\\delta_n,k\\delta_n, u^{[n,k-1]}}),\\;\\; k=2,\\ldots .\\]\n\t\t\t\nNote that by construction, the restriction of $ u^{[n,k]} $ to interval $[0,(k-1)\\delta_n]$ is equal to $u^{[n,k-1]}$.\n\n\n\\item[\\textbf{Step 4.}]\nBy \\textbf{Step 3} we can define a process $u^n\\in \\mathscr{M}^2(X_T)$ by $u^n(t)=u^{[n,k]}(t)$, if $t \\in [0,k\\delta_n]$. Moreover, for every $t\\in [0,T]$, $\\mathbb{P}$-a.s.,\n\t\t\t\\begin{equation}\\label{Eq:truncated-equation}\n\t\t\tu^n(t)=S_t u_0+\\int_0^t S_{t-r}[\\theta_n (|u^n|_{X_r})F(u^n(r))]dr+\\int_0^t\n\t\t\tS_{t-r}[\\theta_n (|u^n|_{X_r})G(u^n(r))]dW(r).\n\t\t\t\\end{equation}\n\n\n\\item[\\textbf{Step 5.}]\nLet $(\\tau_n)_{n\\in \\mathbb{N}}$ be a sequence of stopping\n\t\t\ttimes defined by\n\t\t\t\\begin{equation}\\label{eqn-stopping time tau_n}\n\t\t\t\\tau_n=\\inf\\{t\\in [0,\\infty): |u^n|_{X_t}\\ge n\\}.\n\t\t\t\\end{equation}\t\t\n\t\n\nLet us fix $n\\in \\mathbb{N}$.\n By \\cite[Lemma A.1]{Brz+Masl+Seidler_2005}, we infer from \\eqref{Eq:truncated-equation} that for every $t\\in [0,\\infty)$, $\\mathbb{P}$-a.s.\n\t\t\t\\begin{equation}\\label{Eq:truncated-equation-2}\n\t\t\tu^n(t \\wedge \\tau_n)=S_{t\\wedge \\tau_n} u_0+\\int_0^{t\\wedge \\tau_n} S_{t\\wedge \\tau_n -r}[\\theta_n (|u^n|_{X_r})F(u^n(r))]\\,dr\n+\\tilde{I}^n_{\\tau_n}(t \\wedge \\tau_n),\n\t\t\t\\end{equation}\nwhere $\\tilde{I}^n_{\\tau_n}$ is a continuous $V$-valued\n\\[\n\\tilde{I}^n_{\\tau_n}(t):= \\int_0^t \\mathds{1}_{[0,\\tau_n)}(s)\n\t\t\tS_{t-r}[\\theta_n (|u^n|_{X_r})G(u^n(r))]dW(r),\\;\\; t\\in [0,\\infty).\n\\]\n\tBy the definition of the function $\\theta_n$ we infer that $\\theta_n(|u^n|_{X_{r}})=1$ for $r\\in [0,t\\wedge\n\t\t\t\\tau_n)$. Hence\n\t\t\t$$\\theta_n (|u^n|_{X_r})F(u^n(r)) = F(u^n(r)), r\\in [0,t\\wedge\n\t\t\t\\tau_n), \\; t\\in [0,\\infty).$$\nTherefore, we deduce that for every $t\\in [0,\\infty)$, $\\mathbb{P}$-a.s.\n\\[\n\\tilde{I}^n_{\\tau_n}(t)= \\int_0^t \\mathds{1}_{[0,\\tau_n)}(s)\n\t\t\tS_{t-r}[G(u^n(r))]dW(r)=:I^n_{\\tau_n}(t).\n\\]\nThus, we infer that\n\t\t\t$u^n$ satisfies, for every $t\\in [0,\\infty)$, $\\mathbb{P}$-a.s.\n\t\t\t\\begin{equation}\\label{Eq:truncated-equation-2-A}\n\t\t\tu^n(t \\wedge \\tau_n)=S_{t\\wedge \\tau_n} u_0+\\int_0^{t\\wedge \\tau_n} S_{t\\wedge \\tau_n -r}[F(u^n(r))]\\,dr\n+I^n_{\\tau_n}(t \\wedge \\tau_n).\n\t\t\t\\end{equation}\n\n\n\n\\item[\\textbf{Step 6.}]\\label{lem-increasing stopping times}\n Arguing as in the proof of proof of \\cite[Lemma 5.1]{Brz+Millet_2012} we can show that for every $n \\in \\mathbb{N}$,\n\\begin{equation}\n\\label{eqn-increasing stopping times}\n\\tau_n < \\tau_{n+1} \\;\\;\\;\\; \\mathbb{P}\\text{-a.s.}\n\\end{equation}\nand\n\\begin{equation}\n\\label{eqn-u_n=u_n+1}\nu^n(t)=u^{n+1}(t) \\mbox{ if } t\\in [0,\\tau_n) \\mbox{ and } n \\in \\mathbb{N},\\;\\; \\mathbb{P}\\text{-a.s.}\n\\end{equation}\nBy taking appropriate modifications we can assume that \\eqref{eqn-increasing stopping times} is satisfied on the whole space $\\Omega$.\nHence, the following limit exists\n\n\\begin{equation}\\label{eqn-tau_infty=lim}\n\\tau_\\infty(\\omega)=\\lim_{n\\rightarrow \\infty} \\tau_n(\\omega), \\,\\, \\omega \\in \\Omega.\n\\end{equation}\nSince our probability basis satisfies the usual hypothesis, $\\tau_\\infty$ is an {accessible} stopping time, see \\cite[Proposition 2.3 and Lemma 2.11]{Kar-Shr-96} with $(\\tau_n)$ being the announcing sequence for $\\tau_\\infty$. The two claims made at the beginning of \\textbf{Step 6} enable us to define a local process $(u,\\tau_\\infty)$ in the following way\n\\begin{equation}\n\\label{eqn-local process u}\n u(t,\\omega)=u^n(t,\\omega) \\text{ if } t<\\tau_n(\\omega), \\omega \\in \\Omega.\n\\end{equation}\n\\item[\\textbf{Step 7.}]\\label{Step7} We claim that $(u,\\tau_\\infty)$ is a local solution to problem \\eqref{ABS-SPDE-strong}. \\\\\nIndeed, arguing as in \\textbf{Step 5}, in particular using \\cite[Lemma A.1]{Brz+Masl+Seidler_2005}, we can show that\n\t\t\t$u^n$ satisfies, for every $t\\in [0,\\infty)$, $\\mathbb{P}$-a.s.\n\t\t\t\\begin{equation}\\label{eqn-local-3}\n\t\t\tu(t \\wedge \\tau_n)=S_{t\\wedge \\tau_n} u_0+\\int_0^{t\\wedge \\tau_n} S_{t\\wedge \\tau_n -r}[F(u(r))]\\,dr\n+I_{\\tau_n}(t \\wedge \\tau_n).\n\t\t\t\\end{equation}\t\nwhere $I_{\\tau_n}$ is an $V$-valued continuous process defined by\n\\[\nI_{\\tau_n}(t)= \\int_0^t \\mathds{1}_{[0,\\tau_n)}(s)\n\t\t\tS_{t-r}[G(u(r))]\\,dW(r), \\;\\; t\\in [0,\\infty).\n\\]\nSince, as observed above, $\\tau_\\infty$ is an accessible stopping time with the announcing sequence $(\\tau_n)$, by definition \\ref{def-local solution} we infer that\nthe local\t\t\tprocess $(u, \\tau_\\infty)$ is a local solution to problem \\eqref{ABS-SPDE-strong}.\n\nThis also ends the proof of the first part of Theorem \\ref{thm_local}.\n\t\t\t\\end{trivlist}\n\n\n\\noindent \t\\textit{Proof of part (III)}\n\t\n\tLet us recall that $\\Omega_1 \\in \\mathcal{F}_0$. Let $i:\\Omega_1 \\hookrightarrow \\Omega$ be the natural embedding and $W^1$ a process given by \n\t\\[\n\tW^1(t):=W(t)\\circ i,\\;\\; t\\geq 0.\n\t\\]\n\tWe define $\\mathcal{F}_t^1$ by\n\t\\[\n\t\\mathcal{F}_t^1:=\\bigl\\{ A\\cap \\Omega_1=i^{-1}(A): A \\in \\mathcal{F}_t \\bigr\\}, \\;\\;\\; t\\geq 0.\n\t\\]\n\tSimilarly we define $\\mathcal{F}^1$. We put $\\mathbb{F}^1=\\bigl(\\mathcal{F}_t^1\\bigr)_{t \\geq 0}$.\n\t\n\tWe also define a measure\n\t\\[\n\t\\mathbb{P}^1: \\mathcal{F}^1 \\ni A\\cap \\Omega_1 \\mapsto \\frac{\\mathbb{P}(A\\cap \\Omega_1)}{\\mathbb{P}(\\Omega_1)} \\in [0,1].\n\t\\]\n\tIt is easy to check that\n\t$\n\t\\bigl(\\Omega_1,\\mathcal{F}^1,\\mathbb{P}^1, \\mathbb{F}^1\\bigr)\n\t$\n\tis a filtered probability space satisfying the usual conditions. Moreover, since the Wiener process $W$ is independent of $\\mathcal{F}_0$,\n\t$W^1$ is a Wiener process on $\\bigl(\\Omega_1,\\mathcal{F}^1,\\mathbb{P}^1, \\mathbb{F}^1\\bigr)$.\n\t\n\\noindent\tHence, it is sufficient to prove our result in the case when $\\Omega_1=\\Omega$. Indeed, the following holds true.\n\t\\begin{lem}\\label{lem-localization of the probability space}\n\t\tIf\n\t\ta local process $u=\\big(u(t),\n\t\tt\\in[0, \\tau) \\big) $ is a\n\t\tlocal solution to problem \\eqref{ABS-SPDE-strong} on the original probability basis\n\t\t$\\bigl(\\Omega,\\mathcal{F},\\mathbb{P}, \\mathbb{F}\\bigr)$, then the\n\t\tprocess $u^1=\\big(u(t),\t\tt\\in[0, \\tau^1) \\big) $ defined by \n\t\t\\[\n\t\tu^1(t,\\omega_1):=u(t,i(\\omega_1)), \\;\\; t \\in [0,\\tau(i(\\omega_1)),\\;\\; \\omega_1 \\in \\Omega_1,\n\t\t\\]\n\t\tand $\\tau^1:=\\tau\\circ i$, \n\t\tis a local solution to problem \\eqref{ABS-SPDE-strong} on the new probability basis\n\t\t$\\bigl(\\Omega_1,\\mathcal{F}^1,\\mathbb{P}^1, \\mathbb{F}^1\\bigr)$.\n\t\\end{lem}\n\t\n\tThus we assume that $u_0$ is a $\\mathcal{F}_0$-measurable $V$-valued random variable such that\n\t\\[\\Vert u_0 \\Vert \\leq R \\mbox{ $\\mathbb{P}$-a.s. on } \\Omega.\\]\n\t\n\t\n\t\n\tOur proof follows the lines of \\cite[Theorem 5.3]{Brz+Millet_2012}. In order to simplify the notation we will write $\\Psi^n_{T}$ instead of $\\Psi^n_{0,T,u_0}$, We will also write $\\Phi^n_F$ (resp. $\\hat{\\Phi}^n_G$) instead of $\\Phi^n_{\\delta,T,\\mathbf{a}}$ (resp. $\\hat{\\Phi}^n_{\\delta,T,\\mathbf{a}}$).\n\t\n\n\tWe modify the initial data $u_0$ by replacing it by $\\tilde{u}_0=u_01_{\\Omega_1}$. Then $ \\Vert \\tilde{u}_0 \\Vert\\leq R$ on $\\Omega$.\n\t\n\t\\begin{trivlist}\n\t\t\\item[\\textbf{Step 8.}] Let us fix $\\varepsilon\n\t\t>0$ and choose $M$\n\t\tsuch that $M\\ge (C_0+1) \\varepsilon^{-\\frac 12}$, where $C_0$ is the constant appearing in inequality \\eqref{ineq-C_0} below.\n\t\tThanks to Propositions \\ref{prop-global Lipschitz-F} and \\ref{prop-Psi_T-Lipschitz}, Corollary \\ref{cor-global-lip-G} and Assumption\n\t\t\\ref{assum-01} we can find $\\tilde{D}_i(n)>0, i=1,2$, $n\\in\\mathbb{N}$ such that for all $u\\in \\mathscr{M}^2(\\hat{X}_{0,T,u_0})$ we have\n\t\t\\begin{align*}\n\t\t\\lvert S\\ast \\Phi_F^n(u)\\rvert_{\\mathscr{M}^2(X_T)}\n\t\t\\le \\tilde{D}_1(n) \\tilde{C}_1 \\tilde{C}_2(2n \\tilde{C}_2+1)(2n)^{p+2}T^{(1-\\alpha)\/2},\\\\\n\t\t\\lvert S\\diamond \\hat{\\Phi}^n_G(u)\\rvert_{\\mathscr{M}^2(X_T)}\\le \\tilde{D}_2(n) \\tilde{C}_3\n\t\t\\tilde{C}_4 (2n)^{k+2} [2n \\tilde{C}_4+1] T^{(1-\\beta)\/2}.\n\t\t\\end{align*}\n\t\tHence,\t\tsince $1-\\alpha, 1-\\beta >0$ we infer\n\t\tthat there exists a sequence $(K_n(T))_n$ of numerical functions\n\t\tsuch that for all $n$, $\\lim_{T\\rightarrow 0} K_n(T)=0$ and\n\t\t\\begin{equation*}\n\t\t\\lvert S\\ast \\Phi^n_T(u)+ S\\diamond\n\t\t\\Phi^n_G(u)\\rvert_{\\mathscr{M}^2(X_T)}\\le K_n(T),\\;\\; u\\in \\mathscr{M}^2(X_T).\n\t\t\\end{equation*}\n\t\tLet us put $n=M R^2$ \n\t\tand\n\t\tchoose $\\delta_{1}(\\varepsilon,R) >0$ such that\n\t\t$K_{n}(\\delta_{1}(\\varepsilon,R)) \\leq R $. Let $\\Psi^n_T$ be the\n\t\tmapping defined by \\eqref{eqn-Psi_T}. Since $\\mathbb{E} \\lVert \\tilde{u}_0\\rVert^2\\leq\n\t\tR^2 $, we infer by the Assumption \\ref{assum-01} that\n\t\t\\begin{align} \\label{ineq-C_0}\n\t\t\\lvert \\Psi^n_T(u)\\rvert_{\\mathscr{M}^2(X_T)}\\le & C_0 R + K_n(T)\n\t\t\\le (C_0+ 1)R ,\n\t\t\\end{align}\n\t\tfor all $T\\le \\delta_1(\\varepsilon,R)$. \n\t\tFurthermore, Propositions \\ref{prop-Psi_T-Lipschitz}\n\t\timplies that there exists $C>0$ such that \n\t\t\\begin{equation*}\n\t\t\\lvert \\Psi^n_T(u_1)-\\Psi^n_T(u_2)\\rvert_{\\mathscr{M}^2(X_T)}\\le C_3\n\t\tM^{D}R^{D} \\Big[T^{(1-\\alpha)\/2} \\vee T^{(1-\\beta)\/2}\\Big] \\lvert\n\t\tu_1-u_2\\rvert_{\\mathscr{M}^2(X_T)}, \\text{ for all $u_1,u_2\\in\n\t\t\t\\mathscr{M}^2(X_T)$}.\n\t\t\\end{equation*}\n\t\tSince $M\\ge (C_0+1) \\varepsilon^\\frac 12$ we infer that\n\t\t\\begin{equation*}\n\t\t\\lvert \\Psi^n_T(u_1)-\\Psi^n_T(u_2)\\rvert_{\\mathscr{M}^2(X_T)}\\le C_3\n\t\tM^{D}R^{D} \\Big[T^{(1-\\alpha)\/2} \\vee T^{(1-\\beta)\/2}\\Big] \\lvert\n\t\tu_1-u_2\\rvert_{\\mathscr{M}^2(X_T)}.\n\t\t\\end{equation*}\n\t\t\n\t\t\n\t\tHence we can find $\\delta_2(\\varepsilon,R)>0$ such that $\\Psi^n_T$ is a\n\t\tstrict contraction for all $T\\le \\delta_2(\\varepsilon,R)$. Thus if one\n\t\tputs $T^\\ast(\\varepsilon,R)=\\delta_1(\\varepsilon,R)\\wedge \\delta_2(\\varepsilon,R)$, the\n\t\tmapping $\\Psi^n_T$ has a unique fixed point ${\\hat{u}^n}$ which satisfies\n\t\t\\begin{equation}\\label{ineq-moment-1}\n\t\t\\mathbb{E} \\lvert {\\hat{u}^n}\\rvert^2_{X_{T^\\ast(\\varepsilon,R)}}\\le (C_0+1)^2 R^2.\n\t\t\\end{equation}\n\t\tSimilarly to \\eqref{eqn-stopping time tau_n} we can define a new stopping time ${\\hat{\\tau}_n}$ by\n\t\t\\[\n\t\t{\\hat{\\tau}_n}:= \\inf\\{t\\in [0,\\infty): |{\\hat{u}^n}|_{X_t}\\ge n\\}.\n\t\t\\]\n\t\tArguing as in \\textbf{Step 5} we can show that \t\t\t${\\hat{u}^n}$ satisfies, for every $t\\in [0,T]$, $\\mathbb{P}$-a.s.\n\t\t\\begin{equation}\\label{Eq:truncated-equation-9-1}\n\t\t{\\hat{u}^n}(t \\wedge \\tau_n)=S_{t\\wedge \\tau_n} u_0+\\int_0^{t\\wedge {\\hat{\\tau}_n}} S_{t\\wedge {\\hat{\\tau}_n} -r}[F({\\hat{u}^n}(r))]\\,dr\n\t\t+{\\hat{I}^n_{\\hat{\\tau}_n}}(t \\wedge {\\hat{\\tau}_n}).\n\t\t\\end{equation}\n\t\twhere ${\\hat{I}^n_{\\hat{\\tau}_n}}$ is a continuous $V$-valued process defined by\n\t\t\\[\n\t\t{\\hat{I}^n_{\\hat{\\tau}_n}}(t):=\\int_0^t \\mathds{1}_{[0,\\tau_n)}(s)\n\t\tS_{t-r}[G({\\hat{u}^n}(r))]dW(r), \\;\\; t\\in [0,T].\n\t\t\\]\n\t\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\tBy the definition of the stopping time ${\\hat{\\tau}_n}$,\n\t\t$\\{{\\hat{\\tau}_n} \\le T^\\ast(\\varepsilon,R)\\} \\subset \\{\n\t\t\\lvert u^n\\rvert_{X_{T^\\ast(\\varepsilon,R)}}\\ge n \\}$. Therefore, by the Chebyshev\n\t\tinequality and inequality \\eqref{ineq-moment-1} we infer that \t\t\n\t\t\\begin{align*}\n\t\t\\mathbb{P}({\\hat{\\tau}_n} \\le T^\\ast(\\varepsilon,R))\\le &\\mathbb{P}(\\lvert\n\t\tu^n\\rvert^2_{X_{T^\\ast(\\varepsilon,R)}}\\ge n ) \\leq \\frac{1}{n}\\mathbb{E}\\lvert\n\t\tu^n\\rvert^2_{X_{T^\\ast(\\varepsilon,R)}} \\leq \\frac{1}{n} (C_0+1)^2R^2.\n\t\t\\end{align*}\n\t\tSince $n=NR^2$ and $N\\ge (C_0+1)^2 \\varepsilon^{-\\frac 12}$ we get\n\t\t\\begin{align*}\n\t\t\\mathbb{P}({\\hat{\\tau}_n} \\le T^\\ast(\\varepsilon,R))\\leq & (C_0+1)^2 N^{-2} \\leq \\varepsilon.\n\t\t\\end{align*}\n\t\t\n\t\tHence, we have prove \\eqref{ineq-positive probability-2}.\n\t\t\n\t\t\\item[\\textbf{Step 9.}]\\label{Step9}\n\t\t\n\t\tTo conclude the proof, we observe that in view of Remark \\ref{rem-local solution}, it follows from equality \\eqref{Eq:truncated-equation-9-1} that the process\n\t\t${\\hat{u}^n}$ restricted to the open random interval $[0, \\hat{\\tau}^n)\\times \\Omega$ is a local\n\t\tsolution to problem \\eqref{ABS-SPDE-strong}. On the other hand, in \\textbf{Step 7} we proved that also $(u,\\tau_\\infty)$ is a local solution to problem \\eqref{ABS-SPDE-strong}.\n\t\t{Because local uniqueness holds for problem \\eqref{ABS-SPDE-strong}}, see Theorem \\ref{Thm:LocalUniqueness}, we infer by applying Corollary \\ref{cor-max of two local solutions} that the supremum\n\t\tof $(u,\\tau_\\infty)$ and $({\\hat{u}^n},\\hat{\\tau}^n)$ is another local\n\t\tsolution to problem \\eqref{ABS-SPDE-strong}.\n\t\tHence, the stopping time $T_1= \\tau_\\infty \\vee \\hat{\\tau}^n$ satisfies the requirements of the theorem.\n\t\\end{trivlist}\n\tThis concludes the proof of part (III) of Theorem \\ref{thm_local} and thus of the whole theorem.\n\t\t\n\\end{proof}\n\n The next result is about the existence and uniqueness of a maximal solution and the characterization of its lifespan.\n\t\\begin{thm}\\label{thm_maximal-abstract}\n\tLet $u_0\\in L^2(\\Omega,\\mathcal{F}_0,V)$. Then, problem \\eqref{ABS-SPDE-strong} has a unique maximal local solution $(\\hat{u},\\hat{\\tau})$.\n\t\tMoreover, $(u,\\tau_\\infty) \\sim (\\hat{u},\\hat{\\tau})$ and\n\t\\begin{align}\n\t&\t\\lim_{t\\toup \\hat{\\tau}}|\\hat{u}|_{X_t}=\\infty \\;\\;\\; \\mathbb{P}\\mbox{-a.s. on\n\t\t} \\{\\hat{\\tau}<\\infty\\},\\label{eqn-t_infty}\\\\\n&\t\t\\mathbb{ P}\\Big(\\{\\omega \\in\\Omega :\\hat{\\tau} (\\omega)< \\infty \\text{ and }\n\t\t\\sup_{t\\in [0, \\hat{\\tau} (\\omega))} \\lvert \\hat{u}(t)(\\omega)\\rvert_V<\\infty \\}\\Big)=0. \\label{eqn-t_infty-limit}\n\t\t\\end{align}\n\t\n\t\\end{thm}\n\n\n\t\\begin{proof}\n\t\tLet us choose and fix $u_0\\in L^2(\\Omega,\\mathcal{F}_0,V)$. Wlog we can assume that $\\mathbb{P}(\\{\\hat{\\tau}<\\infty\\})>0$.\n\t\t\n\t\tFirstly, we observe that it follows from the proof of Theorem \\ref{thm_local} that the local process $(u,\\tau_\\infty)$ defined in \\eqref{eqn-local process u} is a local solution to \\eqref{ABS-SPDE-strong}.\n In particular, the set of local solutions to problem \\eqref{ABS-SPDE-strong} is non-empty. Since by Theorem \\ref{Thm:LocalUniqueness} the local uniqueness holds for problem \\eqref{ABS-SPDE-strong}, we infer by applying Proposition \\ref{prop-loc-implies-max} that there exists a unique maximal local solution $(\\hat{u},\\hat{\\tau})$ to \\eqref{ABS-SPDE-strong}. Moreover,\n\t\t$(\\hat{u},\\hat{\\tau})$ satisfies the following\n\t\t\\begin{equation}\\label{eqn-similarity}\n\t\t\\hat{\\tau} \\ge \\tau_\\infty \\text{ $\\mathbb{P}$-a.s. and } \\hat{u}_{\\lvert_{[0,\\tau_\\infty)\\times \\Omega} }= u.\n\t\t\\end{equation}\t\t\nSecondly, suppose that $\\mathbb{P} \\bigl( \\hat{\\tau} > \\tau_\\infty \\bigr)>0$. Let $\\bigl(\\hat{\\tau}_n\\bigr)$ be the announcing sequence of $\\hat{\\tau}$. Since $\\hat{\\tau}_n \\toup \\hat{\\tau}$ $\\mathbb{P}$-a.s., we infer that\nthere exists $n \\in \\mathbb{N}$ such that $\\mathbb{P} \\bigl( \\hat{\\tau} > \\hat{\\tau}_n> \\tau_\\infty \\bigr)>0$. Thus we infer that there exists $t>0$ such that\n$\\mathbb{P} \\bigl( t> \\hat{\\tau} > \\hat{\\tau}_n> \\tau_\\infty \\bigr)>0$. \\\\\nOn the other hand, by the definition \\eqref{eqn-stopping time tau_n} of the announcing sequence $(\\tau_n)_{n \\in \\mathbb{N}}$ for $\\tau_\\infty$ and the definition, see \\eqref{eqn-local process u}, of the local process $(u,\\tau_\\infty)$ that the sequence $\\vert u \\vert_{X_{\\tau_n}}$ converges to $\\infty$ on $\\{ \\tau_\\infty< \\infty\\}$. Hence we infer that $\\lvert u\\rvert_{X_{\\tau_\\infty}} = \\infty $ $\\mathbb{P}$-a.s. on\n $\\bigl\\{ t> \\hat{\\tau} > \\hat{\\tau}_n> \\tau_\\infty \\bigr\\}$. Since the probability of the last set is $>0$, this contradicts condition \\eqref{eq-locsol_00} of Definition\n \\ref{def-local solution-2} of a local solution. This contradiction implies that $\\mathbb{P} \\bigl( \\hat{\\tau} > \\tau_\\infty \\bigr)=0$, and in view of \\eqref{eqn-similarity}, we also have $(u,\\tau_\\infty) \\sim (\\hat{u},\\hat{\\tau})$.\n\n\n\n\n\t\tThirdly, we infer from the definition \\eqref{eqn-stopping time tau_n} of the announcing sequence $(\\tau_n)_{n \\in \\mathbb{N}}$ for $\\tau_\\infty$ and the definition, see \\eqref{eqn-local process u}, of the local process $(u,\\tau_\\infty)$ that the sequence $\\vert u \\vert_{X_{\\tau_n}}$ converges to $\\infty$ on $\\{\\tau_\\infty< \\infty\\}$. Since\nthe function $t \\mapsto \\vert u \\vert_{X_{t}}$ is increasing, by \\eqref{eqn-similarity}, we infer that\n\\begin{equation}\\label{eqn-blow-up similarity}\n\\dela{\\lim_{t\\toup \\hat{\\tau}} \\lvert \\hat{u} \\rvert_{X_t} \\ge\n\\lim_{t\\toup \\tau_\\infty} \\lvert \\hat{u} \\rvert_{X_t} =} \\lim_{t\\toup \\tau_\\infty}\\lvert u \\rvert_{X_t} = \\lim_{n \\toup \\infty} \\lvert u\\rvert_{X_{\\tau_n}} = \\infty \\text{ $\\mathbb{P}$-a.s. on } \\{\\hat{\\tau}< \\infty \\}.\n\\end{equation}\nThis proves\t\\eqref{eqn-t_infty}.\n\nFourthly, we shall prove \\eqref{eqn-t_infty-limit}. By contradiction, we assume that\nthere exists $\\varepsilon>0$ such that\n\\[ {\\mathbb P }\\big(\\{\\tau_\\infty <\\infty\\} \\cap \\{ \\sup_{t\\in [0,\\tau_\\infty)} |\\hat{u}(t)|_{V}<\\infty\n\\}\\big)=4\\varepsilon>0.\\]\nHence we can easily deduce that there exists $R>0$ such that\n\\[ {\\mathbb P }\\big( \\{ |u(t)|_{V}0$ depending only on $\\varepsilon$ and $R$ comes from \npart (II) of Theorem \\ref{thm_local}. By the definition of an announcing sequence \n $\\bigl(\\tau_n\\bigr)$ of $\\tau_\\infty$ we infer that\nfor arbitrary $\\delta>0$ there exists $n_0>0$ such that\n$\\mathbb{P}(\\Omega_0)\\geq (1-\\delta)\\mathbb{P}(\\tilde{\\Omega})$,\nwhere $\\Omega_0:=\\{\\omega \\in \\tilde{\\Omega}: |u(t)|_{V}0$, by applying part (II) of Theorem \\ref{thm_local} we find a local solution $y(t)$, $t\\in\n[\\tau_{n_0},\\tau_{n_0} +T_1)$ to problem \\eqref{ABS-SPDE-1} with the initial condition (starting at $\\tau_{n_0}$) $y(\\tau_{n_0})=y_0$ such that\n$\\mathbb{P}(T_1\\geq T^\\ast(\\varepsilon,R))>1-\\varepsilon$. \nAlso, let $\\Omega_1:= \\Omega_0 \\cap \\{ T_1\\geq T^\\ast(\\varepsilon,R)\\}.$ Since\n$\\mathbb{P}(\\hat{\\tau}-T_0< \\frac 12 T^\\ast(\\varepsilon,R))\\geq 2\\varepsilon$, we infer that\n\\[\\mathbb{P}\\big(\\Omega_1 \\big) \\geq \\varepsilon>0.\\]\n\nBy a generalization of \\cite[Corollary 2.28]{Brz+Elw_2000} to the case of SPDEs we infer that a local stochastic process $v(t)$, $t\\in [0,\\tau_{n_0}(\\omega) +T_1)$ defined by\n\\begin{equation*}\nv(t,\\omega)=\\begin{cases}\nu(t,\\omega) \\text{ if } t \\in [0,\\tau_{n_0}(\\omega)] ,\\\\\ny(t,\\omega) \\text{ if } t\\in [ \\tau_{n_0}(\\omega), \\tau_{n_0}(\\omega) +T_1 ).\\\\\n\\end{cases}\n\\end{equation*}\nis a local solution to problem \\eqref{ABS-SPDE-1} with the initial data $v(0)=u_0$. However, on the $\\Omega_1$, we have \n\\[\n\\tau_{n_0} +T_1-\\tau_\\infty = T_1 -( \\tau_\\infty -\\tau_{n_0} )\\geq 2\\alpha -\\alpha=\\alpha>0,\n\\]\nwhat contradicts the maximality of the solution $(u,\\tau_\\infty)$ proved in the earlier part of the theorem.\nThis completes the proof of \\eqref{eqn-t_infty-limit} as well as the theorem.\n\n\t\\end{proof}\n\t\\subsection*{Acknowledgments}\n\tThis article is part of a project that is currently funded by the European Union's Horizon 2020 research and innovation programme under the Marie Sk\\l{}odowska-Curie grant agreement No. 791735 ``SELEs\".\n\tThe authors are also grateful for the support they received from the FWF-Austrian Science through the Stand-Alone project P28010.\n\tZ. Brze{\\' z}niak presented a\n\tlecture based on a preliminary version of this paper at the RIMS Symposium on Mathematical Analysis of Incompressible Flow held at Kyoto in February 2013.\n\tHe would like to thank Professor Toshiaki Hishida for the kind invitation. Razafimandimby is also very grateful to the organizers of the conference `` Nonlinear PDEs in Micromagnetism: Analysis, Numerics and Applications'', which was held in ICMS Edinburgh, UK, for their invitation to present a talk based on preliminary result of this paper at this meeting. He is very grateful for the financial support he received from the International Centre for Mathematical Sciences (ICMS) Edinburgh.\n\tLast, but not the least, the authors wish to thank Professor Guoli Zhou for pointing out some gaps in the previous version of this paper \\cite{BHP-arxiv}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\\section{The ATLAS detector}\n\\label{sec:atlas}\n\nThe ATLAS experiment~\\cite{ATLASdetector} is a general-purpose detector consisting of an inner tracker, a calorimeter and a muon spectrometer. \nThe inner detector (ID) directly surrounds the interaction point; \nit consists of a silicon pixel detector, a semiconductor tracker and a transition radiation tracker, and is embedded in an axial~$2$ T magnetic field.\nThe ID covers the pseudorapidity\\footnote{ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) \nin the centre of the detector and the $z$-axis along the beam pipe.\nThe $x$-axis points from the IP to the centre of the LHC ring, and the $y$-axis points upward. Cylindrical \ncoordinates $(r,\\phi)$ are used in the transverse plane,\n$\\phi$ being the azimuthal angle around the beam pipe. The pseudorapidity $\\eta$ is defined in terms of the \npolar angle $\\theta$ as $\\eta=-\\ln\\tan(\\theta\/2)$ and the\ntransverse momentum $p_{\\rm T}$ is defined as $p_{\\rm T}=p\\sin\\theta$. The rapidity is defined as \n$y=0.5\\ln\\left[\\left( E + p_z \\right)\/ \\left( E - p_z \\right)\\right]$,\nwhere $E$ and $p_z$ refer to energy and longitudinal momentum, respectively. The $\\eta$--$\\phi$ distance \nbetween two particles is defined as\n $\\Delta R=\\sqrt{(\\Delta\\eta)^2 + (\\Delta\\phi)^2}$.}\nrange $|\\eta| = $ 2.5 and is enclosed by a calorimeter system containing\nelectromagnetic and hadronic sections.\nThe calorimeter is surrounded by a large muon spectrometer (MS) in a toroidal magnet system.\nThe MS consists of monitored drift tubes and cathode strip chambers, designed to provide\nprecise position measurements in the bending plane in the range $|\\eta| <$ 2.7.\nMomentum measurements in the muon spectrometer are based on track segments formed in at least two of the three precision chamber planes.\n\nThe ATLAS trigger system \\cite{ATLAS:trig} is separated into three levels: the hardware-based Level-1 trigger\nand the two-stage High Level Trigger (HLT), comprising the Level-2 trigger and Event\nFilter, which reduce the 20~MHz proton--proton collision rate to several-hundred Hz of events of interest for data recording to mass storage. \nAt Level-1, the muon trigger searches for patterns of hits satisfying different transverse momentum thresholds with a coarse \nposition resolution but a fast response time using resistive-plate chambers and thin-gap chambers in the ranges $|\\eta| <$ 1.05 and $1.05 <|\\eta| < 2.4$, respectively.\nAround these Level-1 hit patterns ``Regions-of-Interest'' (RoI) are defined that\nserve as seeds for the HLT muon reconstruction. \nThe HLT uses dedicated algorithms to incorporate information from both the \nMS and the ID, achieving position and momentum resolution close to that provided by the offline muon reconstruction.\n\n\n\n\\section{Summary and conclusions}\n\\label{sec:conclusion}\n\nThe prompt and non-prompt production cross-sections, the non-prompt production fraction of the $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$\ndecaying into two muons, the ratio of prompt $\\ensuremath{\\psi(2\\mathrm{S})}$ to prompt $\\jpsi$ production, and the ratio of non-prompt $\\ensuremath{\\psi(2\\mathrm{S})}$ to non-prompt $\\jpsi$ production\nwere measured in the rapidity range\n$|y|<2.0$ for transverse momenta between $8$ and $110$~\\GeV. This measurement was carried out using \\mbox{$2.1\\ifb$} (\\mbox{$11.4\\ifb$}) of $pp$\ncollision data at a centre-of-mass energy of $7$ \\TeV\\ ($8$ \\TeV) recorded by the ATLAS experiment at the LHC. It is the latest in a series of \nrelated measurements of the production of charmonium states\nmade by ATLAS.\nIn line with previous measurements, the central values were obtained \nassuming isotropic $\\psi \\to \\mu\\mu$ decays.\nCorrection factors for these cross-sections, computed for a number of extreme spin-alignment scenarios, are between $-35\\%$ and $+100\\%$ at the lowest transverse momenta studied, and between $-14\\%$ and $+9\\%$ at the highest transverse momenta, depending on the specific scenario.\n\nThe ATLAS measurements presented here extend the range of existing measurements to higher transverse momenta, and to a higher collision energy of $\\sqrt{s} = 8$ \\TeV, and, in overlapping phase-space regions, are consistent with previous measurements made by ATLAS and other LHC experiments.\nFor the prompt production mechanism, the predictions from the NRQCD model, which includes colour-octet contributions with various matrix elements \ntuned to earlier collider data, are found to be in good agreement with the observed data points. \nFor the non-prompt production, the fixed-order next-to-leading-logarithm calculations reproduce the data reasonably well, with a slight overestimation of the differential cross-sections at the highest \ntransverse momenta reached in this analysis.\n\n\\section{Candidate selection}\n\\label{sec:data}\n\nThe analysis is based on data recorded at the LHC in 2011 and 2012 during proton--proton collisions at centre-of-mass energies of $7$~\\TeV\\ and $8$~\\TeV, respectively.\nThis data sample corresponds to a total integrated luminosity of \\mbox{$2.1\\ifb$}\\ and \\mbox{$11.4\\ifb$}\\ for $7$~\\TeV\\ data and $8$~\\TeV\\ data, respectively.\n\nEvents were selected using a trigger requiring two oppositely charged muon candidates, each passing the requirement $\\mbox{$p_{\\text{T}}$}>4$~\\GeV. \nThe muons are constrained to originate from a common vertex, which is fitted with the track parameter uncertainties taken into account. \nThe fit is required to satisfy $\\chi^2 < 20$ for the one degree of freedom.\n\nFor $7$~\\TeV\\ data, the Level-1 trigger required only spatial coincidences in the MS~\\cite{ATLAS:2010kba}. For $8$~\\TeV\\ data, a $4$~\\GeV\\ muon $\\mbox{$p_{\\text{T}}$}$ threshold was also applied at Level-1, which reduced the trigger efficiency for low-$\\mbox{$p_{\\text{T}}$}$ muons.\n\nThe offline analysis requires events to have at least two muons, identified by the muon spectrometer and with matching tracks reconstructed in the ID~\\cite{muons}.\nDue to the ID acceptance, muon reconstruction is possible only for $|\\eta| <$ 2.5. \nThe selected muons are further restricted to $|\\eta| <$ 2.3 to ensure high-quality tracking and triggering, and to reduce the contribution from misidentified muons.\nFor the momenta of interest in this analysis (corresponding to muons with a transverse momentum of at most $O(100)$ \\GeV), measurements of the\nmuons are degraded by multiple scattering within the MS and so only the ID tracking\ninformation is considered. \nTo ensure accurate ID measurements, each muon track must fulfil muon reconstruction and selection requirements~\\cite{muons}.\nThe pairs of muon candidates satisfying these quality criteria are required to have opposite charges.\n\nIn order to allow an accurate correction for trigger inefficiencies, each reconstructed muon candidate is required to match a trigger-identified muon\ncandidate within a cone of $\\Delta R = \\sqrt{(\\Delta\\eta)^2 + (\\Delta\\phi)^2}=0.01$.\nDimuon candidates are obtained from muon pairs, constrained to originate from a common vertex using ID track parameters and uncertainties, with a \nrequirement of $\\chi^2 < 20$ of the vertex fit for the one degree of freedom.\nAll dimuon candidates with an invariant mass\nwithin $2.6 < m(\\mu\\mu) < 4.0$ \\GeV\\ and within the kinematic range $\\mbox{$p_{\\text{T}}$}(\\mu\\mu) > 8$ \\GeV, $|y(\\mu\\mu)| < 2.0$ are retained for the analysis.\nIf multiple candidates are found in an event (occurring in approximately $10^{-6}$ of selected events), all candidates are retained.\nThe properties of the dimuon system, such as invariant mass $m(\\mu\\mu)$, transverse momentum $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$, and rapidity $|y(\\mu\\mu)|$ are determined from the result of the vertex fit.\n\n\n\n\\section{Introduction}\n\\label{intro}\n\nMeasurements of heavy quark--antiquark bound states (quarkonia) production processes provide \nan insight into the nature of quantum\nchromodynamics (QCD) close to the boundary between the perturbative and non-perturbative regimes. \nMore than forty years since the discovery of the $\\jpsi$, the investigation of\nhidden heavy-flavour production in hadronic collisions \nstill presents significant challenges to both theory and experiment.\n\nIn high-energy hadronic collisions, charmonium states can be produced either directly by \nshort-lived QCD sources (``prompt'' production), or by long-lived sources in the decay chains of beauty hadrons (``non-prompt'' production). \nThese can be separated experimentally using the\ndistance between the proton--proton primary interaction and the decay vertex of the quarkonium state.\nWhile {\\em Fixed-Order with Next-to-Leading-Log} (FONLL) calculations~\\cite{FONLL_2001,Cacciari:2012ny}, made within the framework of perturbative QCD, have\nbeen quite successful in describing non-prompt production of various quarkonium states, a satisfactory understanding of the prompt production mechanisms is still to be achieved.\n\n\nThe $\\ensuremath{\\psi(2\\mathrm{S})}$ meson is the only vector charmonium state that \nis produced with no significant contributions from decays of higher-mass quarkonia, \nreferred to as feed-down contributions. This\nprovides a unique opportunity to study production mechanisms specific to\n$J^{PC}=1^{--}$ states~\\cite{CDFjpsianomaly1,Abulencia:2007us,Abelev:2014qha,Chatrchyan:2011kc,Chatrchyan:2013cla,Aaij:2012ag,Aad:2014fpa,Aaij:2013jxj,Khachatryan:2015rra,Aaij:2013yaa}.\nMeasurements of the production of $J^{++}$ states with $J=0, 1, 2$,\n\\cite{CDFjpsianomaly2,ATLAS:2014ala,Chatrchyan:2012ub,Aaij:2013yaa,Aaij:2013dja,LHCb:2012ac},\nstrongly coupled to the two-gluon channel, allow similar studies in the $CP$-even sector, complementary to the $CP$-odd vector sector. \nProduction of $\\jpsi$ mesons\n\\cite{Aad:2011sp,Abulencia:2007us,Aaij:2011jh,Chatrchyan:2011kc,Abelev:2012gx,Chatrchyan:2013cla,Aaij:2013jxj,Abelev:2014qha,CDFjpsianomaly3,CDFjpsianomaly1,D0jpsi1,D0jpsi2,Khachatryan:2015rra,Aad:2014fpa,CDFjpsianomaly2,LHCb:2012af}\narises from a mixture of different sources, receiving contributions from the production of $1^{--}$ and $J^{++}$ states in comparable amounts.\n\n\nEarly attempts to describe the formation of charmonium\n\\cite{CSM1,CSM2,CSM3,CSM4,CSM5,CSM7,CEM1,CEM2}\nusing leading-order perturbative QCD \ngave rise to a variety of models, none of which could explain the\nlarge production cross-sections measured at the Tevatron\n\\cite{CDFjpsianomaly1,CDFjpsianomaly2,CDFjpsianomaly3,D0jpsi1,D0jpsi2}.\nWithin the colour-singlet model (CSM) \\cite{Lansberg:2008gk}, next-to-next-to-leading-order (NNLO) contributions to the hadronic production of S-wave\nquarkonia were calculated without introducing any new phenomenological parameters. However,\ntechnical difficulties have so far made it impossible to perform the full NNLO calculation, or to \nextend those calculations to the P-wave states. \nSo it is not entirely surprising that the predictions of the model underestimate the experimental data for inclusive production of\n\\jpsi\\ and $\\Upsilon$ states, where the feed-down is significant, but offer a better description for \\ensuremath{\\psi(2\\mathrm{S})}\\ production \\cite{Aad:2011sp,Aad2012dlq}.\n\nNon-relativistic QCD (NRQCD) calculations that include colour-octet (CO) contributions \n\\cite{Bodwin:1994jh} introduce a number of phenomenological parameters --- long-distance matrix elements\n(LDMEs) --- which are determined from fits to the experimental data, and can hence\ndescribe the cross-sections and differential spectra satisfactorily \n{\\cite{CO_LDME1}}. \nHowever, the attempts to describe the polarization of S-wave quarkonium states using this approach\nhave not been so successful~\\cite{Gong:2012ug},\nprompting a suggestion~\\cite{Faccioli:2014cqa} that a more coherent approach is needed for the treatment of polarization\nwithin the QCD-motivated models of quarkonium production. \n\nNeither the CSM nor the NRQCD model gives a satisfactory explanation for the measurement of prompt\n\\jpsi\\ production in association with the $W$~\\cite{Aad:2014rua} and $Z$~\\cite{Aad:2014kba} bosons: in both cases,\nthe measured differential cross-section is larger than theoretical expectations~\\cite{Li:2010hc,Lansberg:2013wva,Gong:2012ah, Mao:2011kf}.\nIt is therefore important to broaden the scope of comparisons \nbetween theory and experiment by providing a variety of experimental information\nabout quarkonium production across a wider kinematic range.\nIn this context, ATLAS has measured the inclusive differential cross-section of \\jpsi\\ production, with $2.3$~pb$^{-1}$ of integrated luminosity~\\cite{Aad:2011sp}, \nat $\\sqrt{s} = 7$~\\TeV\\ using the data collected in 2010, \nas well as the differential cross-sections of the production of $\\chi_c$ states~(4.5~fb$^{-1}$)~\\cite{ATLAS:2014ala}, and of the \\ensuremath{\\psi(2\\mathrm{S})}\\ in its $J\/\\psi\\pi\\pi$ decay mode (2.1 fb$^{-1})$~\\cite{Aad:2014fpa}, at $\\sqrt{s} = 7$~\\TeV\\ with data collected in 2011.\nThe cross-section and polarization measurements \nfrom CDF~\\cite{Abulencia:2007us},\nCMS~\\cite{Chatrchyan:2013cla,Khachatryan:2010yr,CMS},\nLHCb~\\cite{Aaij:2012ag,Aaij:2013jxj,Aaij:2013yaa,Aaij:2012asz,Aaij:2013nlm,Aaij:2014qea} \nand ALICE~\\cite{Abelev:2014qha,Abelev:2011md,Aamodt:2011gj},\ncover a considerable variety of charmonium production characteristics in a wide \nkinematic range (transverse momentum $\\mbox{$p_{\\text{T}}$}\\leq 100$ \\GeV\\ and rapidities $|y|<5$), thus providing a wealth of\ninformation for a new generation of theoretical models.\n\nThis paper presents a precise measurement of\n\\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ production in the dimuon decay mode, both at $\\sqrt{s} = 7$~TeV\nand at $\\sqrt{s} = 8$~TeV.\nIt is presented as a double-differential measurement in transverse momentum and rapidity of the quarkonium state, \nseparated into prompt and non-prompt contributions,\ncovering a range of transverse momenta $8 < \\mbox{$p_{\\text{T}}$}\\leq 110$ \\GeV\\ and rapidities $|y|<2.0$. \nThe ratios of $\\ensuremath{\\psi(2\\mathrm{S})}$ to $\\jpsi$ cross-sections for prompt and non-prompt processes are also reported, as well as the \nnon-prompt fractions of $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$.\n\\section{Methodology}\n\\label{sec:method}\n\nThe measurements are performed in intervals of dimuon $\\mbox{$p_{\\text{T}}$}$ and absolute value of the rapidity ($|y|$). \nThe term ``prompt'' refers to the $\\jpsi$ or $\\ensuremath{\\psi(2\\mathrm{S})}$ states --- hereafter called $\\ensuremath{\\psi}$ to refer to either --- are produced \nfrom short-lived QCD decays, including feed-down from other charmonium states as long as they are also produced\nfrom short-lived sources. If the decay chain producing a $\\ensuremath{\\psi}$ state includes long-lived particles such as\n$b$-hadrons, then such $\\ensuremath{\\psi}$ mesons are labelled as ``non-prompt''. Using a simultaneous fit to the invariant mass of the dimuon and its ``pseudo-proper decay time'' \n(described below), prompt and non-prompt signal and background contributions can be extracted from the data.\n\n\nThe probability for the decay of a particle as a function of proper decay time $t$ follows an exponential distribution, \n$p(t) = 1\/\\tau_{B}\\cdot e^{-t\/\\tau_{B}}$ where $\\tau_{B}$ is the mean lifetime of the particle. For each decay, the proper decay time \ncan be calculated as $t = L m\/p$,\nwhere $L$ is the distance between the particle production and decay vertices, $p$ is the momentum of the particle, and $m$ is its invariant mass.\nAs the reconstruction of non-prompt $\\ensuremath{\\psi}$ mesons, such as $b$-hadrons, does not fully describe the properties of the parent,\nthe transverse momentum of the \ndimuon system and the reconstructed dimuon invariant mass are used to construct the\n``pseudo-proper decay time'', $\\tau = L_{xy} m(\\mu\\mu)\/\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$, where $L_{xy} \\equiv \\vec{L} \\cdot \\vec{\\mbox{$p_{\\text{T}}$}}(\\mu\\mu)\/\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ \nis the signed projection of the distance of the dimuon decay vertex \nfrom the primary vertex, $\\vec{L}$, onto its transverse momentum, $\\vec{\\mbox{$p_{\\text{T}}$}}(\\mu\\mu)$.\nThis is a good approximation of using the parent $b$-hadron information when the $\\psi$ and parent momenta are closely aligned, which is the case for the values of $\\psi$\ntransverse momenta considered here, and $\\tau$\ntherefore can be used to distinguish statistically between the non-prompt and prompt processes (in which the latter are assumed to decay with vanishingly small lifetime).\nIf the event contains multiple primary vertices~\\cite{ATLASdetector}, the primary vertex closest in $z$ to the dimuon decay vertex is selected.\nThe effect of selecting an incorrect vertex has been shown~\\cite{BJpsiPhi} to have a negligible impact on the extraction of prompt and non-prompt contributions.\nIf any of the muons in the dimuon candidate contributes to the construction of the primary vertex, the corresponding tracks are removed and the vertex is refitted.\n\n\n\\subsection{Double differential cross-section determination}\n\\label{sec:s:diff_xSecDet}\n\nThe double differential dimuon prompt and non-prompt production cross-sections times branching ratio \nare measured separately for \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ mesons according to the equations:\n\n\\begin{equation}\n\\frac{\\mathrm{d}^2\\sigma(pp \\rightarrow \\psi)}{\\mathrm{d}\\mbox{$p_{\\text{T}}$}\\mathrm{d}y} \\times \\mathcal{B} (\\psi \\rightarrow \\mu^+\\mu^- ) = \\frac{N_{\\psi}^{\\mathrm{p}}}{\\Delta \\mbox{$p_{\\text{T}}$} \\Delta y \\times \\int\\mathcal{L} \\mathrm{d}t},\n\\label{equ:xSecP}\n\\end{equation}\n\\begin{equation}\n\\frac{\\mathrm{d}^2\\sigma(pp \\rightarrow b\\bar{b} \\rightarrow \\psi)}{\\mathrm{d}\\mbox{$p_{\\text{T}}$}\\mathrm{d}y} \\times \\mathcal{B} (\\psi \\rightarrow \\mu^+\\mu^- ) = \\frac{N_{\\psi}^{\\mathrm{np}}}{\\Delta \\mbox{$p_{\\text{T}}$} \\Delta y \\times \\int\\mathcal{L} \\mathrm{d}t},\n\\label{equ:xSecNP}\n\\end{equation}\n\n\\noindent where $\\int\\mathcal{L} dt$ is the integrated luminosity, $\\Delta \\mbox{$p_{\\text{T}}$} $ and $ \\Delta y$ are the interval sizes in terms of dimuon transverse momentum and\nrapidity, respectively, and $N_{\\psi}^{\\mathrm{p(np)}}$ is the number of observed prompt (non-prompt) $\\psi$ mesons in\nthe slice under study, corrected for acceptance, trigger and reconstruction efficiencies. \nThe intervals in $\\Delta y$ combine the data from negative and positive rapidities.\n\nThe determination of the cross-sections proceeds in several steps. First, a weight is determined for each\nselected dimuon candidate equal to the inverse of the total efficiency for each candidate. \nThe total weight, $w_\\mathrm{tot}$, for each dimuon candidate includes three factors: the fraction of produced $\\psi \\rightarrow \\mu^+\\mu^-$ decays with both\nmuons in the fiducial region $\\mbox{$p_{\\text{T}}$} (\\mu) > 4$ \\GeV\\ and $|\\eta(\\mu)| <$ 2.3 (defined as acceptance, $\\mathcal{A}$), the probability that a candidate\nwithin the acceptance satisfies the offline reconstruction selection ($\\epsilon_\\mathrm{reco}$), and \nthe probability that a reconstructed event satisfies the trigger selection \n($\\epsilon_\\mathrm{trig}$). The weight assigned to a given candidate when calculating the cross-sections is therefore given by:\n\n\\begin{equation*}\nw_{\\mathrm{tot}}^{-1} = \\mathcal{A} \\cdot \\epsilon_{\\mathrm{reco}} \\cdot \\epsilon_{\\mathrm{trig}}.\n\\end{equation*}\n\nAfter the weight determination, an unbinned maximum-likelihood fit is performed to these weighted events in each ($\\mbox{$p_{\\text{T}}$} (\\mu\\mu), \\ |y(\\mu\\mu)|$) interval using the \ndimuon invariant mass, $m(\\mu\\mu)$, and pseudo-proper decay time, $\\tau(\\mu\\mu)$, observables. \nThe fitted yields of $\\jpsi \\rightarrow \\mu^+\\mu^-$ and $\\ensuremath{\\psi(2\\mathrm{S})} \\rightarrow \\mu^+\\mu^-$ are determined separately for prompt and non-prompt processes. \nFinally, the differential cross-section times the $\\psi \\rightarrow \\mu^+\\mu^-$ branching fraction is calculated for each\nstate by including the integrated luminosity and the $\\mbox{$p_{\\text{T}}$}$ and rapidity interval widths as shown in Eqs. (\\ref{equ:xSecP}) and (\\ref{equ:xSecNP}).\n\n\n\\subsection{Non-prompt fraction}\n\\label{sec:s:NPFDet}\n\nThe non-prompt fraction $f_{b}^{\\psi}$ is defined as the number of non-prompt $\\psi$ (produced\nvia the decay of a $b$-hadron) divided by the number of inclusively produced $\\psi$ decaying to muon pairs after applying weighting corrections:\n\n\\begin{equation*}\nf_{b}^{\\psi} \\equiv \\frac{pp \\rightarrow b + X \\rightarrow \\psi + X'}{pp \\xrightarrow{\\mathrm{Inclusive}} \\psi + X'} = \\frac{N^{\\mathrm{np}}_{\\psi}}{N^{\\mathrm{np}}_{\\psi} + N^{\\mathrm{p}}_{\\psi}}, \n\\end{equation*}\n\n\\noindent where this fraction is determined separately for \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}. \nDetermining the fraction from this ratio is advantageous since acceptance and efficiencies largely cancel and the systematic uncertainty is reduced.\n\n\\subsection{Ratio of \\ensuremath{\\psi(2\\mathrm{S})}\\ to \\jpsi\\ production}\n\\label{sec:s:PNPRatioDet}\n\nThe ratio of \\ensuremath{\\psi(2\\mathrm{S})}\\ to \\jpsi\\ production, in their dimuon decay modes, is defined as:\n\n\\begin{equation*}\nR^{\\mathrm{p(np)}} =\\frac{N^{\\mathrm{p(np)}} _{\\psi(2\\mathrm{S})}}{N^{\\mathrm{p(np)}} _{J\/\\psi}},\n\\end{equation*}\n\n\\noindent where $N_{\\psi}^{\\mathrm{p(np)}}$ is the number of prompt (non-prompt) \\jpsi\\ or \\ensuremath{\\psi(2\\mathrm{S})}\\ mesons decaying into a \nmuon pair in an interval of $\\mbox{$p_{\\text{T}}$}$ and $y$, corrected for selection efficiencies and acceptance.\n\nFor the ratio measurements, similarly to the non-prompt fraction, the acceptance and efficiency corrections \nlargely cancel, thus allowing a more precise measurement.\nThe theoretical uncertainties on such ratios are also smaller, as several dependencies, such as\nparton distribution functions and $b$-hadron production spectra, largely cancel in the ratio.\n\n\\subsection{Acceptance}\n \nThe kinematic acceptance $\\mathcal{A}$ for a $\\psi \\rightarrow \\mu^+\\mu^-$ decay with $\\mbox{$p_{\\text{T}}$}$ and $y$ is given by the \nprobability that both muons pass the fiducial selection ($\\mbox{$p_{\\text{T}}$}(\\mu)>4$ \\GeV\\ and $|\\eta(\\mu)|<2.3$).\nThis is calculated using generator-level ``accept-reject'' simulations, based on the analytic formula described below. \nDetector-level corrections, such as bin migration effects due to detector resolution, \nare found to be small. They are applied to the results and are also considered as part of the systematic uncertainties. \n\nThe acceptance $\\mathcal{A}$ depends on five independent variables (the two muon momenta are constrained by the\n$m(\\mu\\mu)$ mass condition), chosen as the $\\mbox{$p_{\\text{T}}$}$, $|y|$ and azimuthal angle $\\phi$ of the $\\psi$ meson in the laboratory frame,\nand two angles\ncharacterizing the $\\psi \\rightarrow \\mu^+\\mu^-$ decay, $\\theta^{\\star}$ and $\\phi^{\\star}$, described in detail in Ref. \\cite{Faccioli:2010kd}.\nThe angle $\\theta^{\\star}$ is the angle between the direction of the positive-muon momentum in the $\\psi$ rest frame\nand the momentum of the $\\psi$ in the laboratory frame, while $\\phi^{\\star}$ is defined as the angle between the dimuon \nproduction and decay planes in the laboratory frame. \nThe $\\psi$ production plane is defined by the momentum of the $\\psi$ in the laboratory frame and the positive $z$-axis direction.\nThe distributions in $\\theta^{\\star}$ and $\\phi^{\\star}$ \ndiffer for various possible spin-alignment scenarios of the dimuon system.\n\nThe spin-alignment of the $\\psi$ may vary depending on the production mechanism, which in turn affects the angular distribution of the dimuon decay.\nPredictions of various theoretical models are quite contradictory, while the recent experimental measurements~\\cite{Chatrchyan:2013cla} indicate that the angular dependence of $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$\ndecays is consistent with being isotropic. \n\nThe coefficients $\\lambda_{\\theta}, \\lambda_{\\phi}$ and~$\\lambda_{\\theta\\phi}$ in\n\\begin{equation}\n\\frac{\\mathrm{d}^2N}{\\mathrm{d}\\cos\\theta^{\\star}\\mathrm{d}\\phi^{\\star}} \\propto 1 + \\lambda_{\\theta} \\cos^2\\theta^{\\star} + \\lambda_{\\phi} \\sin^2\\theta^{\\star}\\cos2\\phi^{\\star} + \\lambda_{\\theta\\phi} \\sin2\\theta^{\\star}\\cos\\phi^{\\star}\n\\label{equ:acc}\n\\end{equation}\n\\noindent are related to the spin-density matrix elements of the dimuon spin wave function. \n\nSince the polarization of the $\\psi$ state may affect acceptance, seven extreme cases that lead to the largest possible variations of \nacceptance within the phase space of this measurement are identified.\nThese cases, described in Table~\\ref{tab:spin}, are used to define a range in which the results may vary under any physically allowed spin-alignment assumptions.\nThe same technique has also been used in other measurements~\\cite{Aad:2014fpa,ATLAS:2014ala,Aad2012dlq}.\nThis analysis adopts the isotropic distribution in both $\\cos\\theta^{\\star}$ and $\\phi^{\\star}$ as nominal,\nand the variation of the results for a number of extreme spin-alignment scenarios is studied and presented as sets of correction factors, detailed further in Appendix~\\ref{sec:spincorrection}.\n \n\\begin{table}[htbp]\n\\begin{center}\n\\caption{Values of angular coefficients describing the considered spin-alignment scenarios.}\n\\vspace{2mm}\n\\begin{tabular}[h]{r|ccc}\n\\hline\\hline\n & \\multicolumn{3}{c}{Angular coefficients} \\\\ \n & $\\lambda_{\\theta}$ & $\\lambda_{\\phi}$ & $\\lambda_{\\theta\\phi}$ \\\\ \\hline\nIsotropic {\\em (central value)} & $0$ & $0$ & $0$ \\\\\nLongitudinal & $-1$ & $0$ & $0$ \\\\\nTransverse positive & $+1$ & $+1$ & $0$ \\\\\nTransverse zero & $+1$ & $0$ & $0$ \\\\\nTransverse negative & $+1$ & $-1$ & $0$ \\\\\nOff-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane positive & $0$ & $0$ & $+0.5$ \\\\\nOff-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane negative & $0$ & $0$ & $-0.5$ \\\\\n\\hline\\hline\n\\end{tabular}\n\\label{tab:spin}\n\\end{center}\n\\end{table}\n\n\n \nFor each of the two mass-points (corresponding to the \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ masses), two-dimensional maps are produced\nas a function of dimuon $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ and $|y(\\mu\\mu)|$ for the set of spin-alignment hypotheses. \nEach point on the map is determined from a uniform sampling over~$\\phi^{\\star}$ and~$\\cos\\theta^{\\star}$, \naccepting those trials that pass the fiducial selections. To account for various spin-alignment scenarios, all trials are weighted according to Eq.~\\ref{equ:acc}.\nAcceptance maps are defined within the range $8 < \\mbox{$p_{\\text{T}}$}(\\mu\\mu) < 110$~\\GeV and $|y(\\mu\\mu)| < 2.0$, corresponding to the data considered in the analysis. \nThe map is defined by 100 slices in $|y(\\mu\\mu)|$ and 4400 in $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$, using 200k trials for each point, resulting in sufficiently high precision that the statistical uncertainty can be neglected.\nDue to the contributions of background, and the detector resolution of the signal, the acceptance for each candidate is determined from a linear interpolation of the two maps, which are generated for the \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ known masses, as a function of the reconstructed mass $m(\\mu\\mu)$.\n\nFigure~\\ref{fig:accMapUnpolBoth} shows the acceptance, projected in $\\mbox{$p_{\\text{T}}$}$ for all the spin-alignment hypotheses for the \\jpsi\\ meson.\nThe differences between the acceptance of the $\\ensuremath{\\psi(2\\mathrm{S})}$ and $\\jpsi$ meson, are independent of rapidity, except near $|y|\\approx2$\nat low $\\mbox{$p_{\\text{T}}$}$. Similarly, the only dependence on $\\mbox{$p_{\\text{T}}$}$ is found below $\\mbox{$p_{\\text{T}}$}\\approx9$~\\GeV.\nThe correction factors (as given in Appendix.~\\ref{sec:spincorrection}) vary most at low $\\mbox{$p_{\\text{T}}$}$, ranging from $-35\\%$ under longitudinal, to $+100\\%$ for transverse-positive scenarios.\nAt high $\\mbox{$p_{\\text{T}}$}$, the range is between $-14\\%$ for longitudinal, and $+9\\%$ for transverse-positive scenarios.\nFor the fraction and ratio measurements, the correction factor is determined from the appropriate ratio of the individual correction factors. \n\n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=0.8\\textwidth]{c_JpsiAcc_All1D_logx.eps}\n \\caption{Projections of the acceptance as a function of $\\mbox{$p_{\\text{T}}$}$ for the \\jpsi\\ meson for various spin-alignment hypotheses.}\n \\label{fig:accMapUnpolBoth}\n \\end{center}\n\\end{figure}\n\n\n\n\\subsection{Muon reconstruction and trigger efficiency determination}\n\n\\noindent The technique for correcting the 7~\\TeV\\ data for trigger and reconstruction inefficiencies is described in detail in Ref.~\\cite{Aad:2014fpa,Aad2012dlq}.\nFor the 8~\\TeV\\ data, a similar technique is used, however different efficiency maps are required for each set of data, and the 8~\\TeV\\ corrections are detailed briefly below.\n\nThe single-muon reconstruction efficiency is determined from a tag-and-probe study in dimuon decays~\\cite{Aad:2014kba}.\nThe efficiency map is calculated as a function of $\\mbox{$p_{\\text{T}}$} (\\mu)$ and $q\\times \\eta(\\mu)$, where $q=\\pm1$ is the electrical charge of the muon, expressed in units of $e$.\n\nThe trigger efficiency correction consists of two components. The first part represents the trigger efficiency for a \nsingle muon in intervals of $\\mbox{$p_{\\text{T}}$} (\\mu)$ and $q\\times \\eta(\\mu)$.\nFor the dimuon system there is a second correction to account for reductions in efficiency due to closely spaced\nmuons firing only a single RoI, \nvertex-quality cuts, and opposite-sign requirements.\nThis correction is performed in three rapidity intervals: 0--1.0, 1.0--1.2 and 1.2--2.3. The correction is a function of $\\Delta R(\\mu\\mu)$ \nin the first two rapidity intervals and a function of $\\Delta R(\\mu\\mu)$ and $|y(\\mu\\mu)|$ in the last interval.\n\nThe combination of the two components (single-muon efficiency map and dimuon corrections) is illustrated \nin Figure~\\ref{fig:EFmu4DataJpsiTandPCmumupolyL2StarBMaxpt50} by plotting the average trigger-weight correction for the events in this analysis\nin terms of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ and $|y(\\mu\\mu)|$.\nThe increased weight at low $\\mbox{$p_{\\text{T}}$}$ and $|y|\\approx 1.25$ is caused by the geometrical acceptance of the muon trigger system and the turn-on threshold behaviour of the muon trigger. At high $\\mbox{$p_{\\text{T}}$}$ the weight is increased due to the reduced opening angle between the two muons.\n\n\n\\begin{figure} [!ht]\n \\begin{center}\n \\includegraphics[scale=0.6]{averageTrigWeightMean.eps}\n \\caption{Average dimuon trigger-weight in the intervals of $\\mbox{$p_{\\text{T}}$} (\\mu\\mu) $ and $ |y(\\mu\\mu)|$ studied in this set of measurements. \n }\n \\label{fig:EFmu4DataJpsiTandPCmumupolyL2StarBMaxpt50}\n \\end{center}\n\\end{figure}\n\n\n\\subsection{Fitting technique}\n\\label{sec:method:fit}\nTo extract the corrected yields of prompt and non-prompt \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ mesons, two-dimensional\nweighted unbinned maximum-likelihood fits are performed on the dimuon invariant mass, $m(\\mu\\mu)$,\nand pseudo-proper decay time, $\\tau(\\mu\\mu)$, in intervals of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ and~$|y(\\mu\\mu)|$.\nEach interval is fitted independently from all the others.\nIn $m(\\mu\\mu)$, signal processes of $\\ensuremath{\\psi}$ meson decays are statistically distinguished as narrow peaks convolved with the detector resolution, at their respective mass positions, on top of background continuum.\nIn $\\tau(\\mu\\mu)$, decays originating with zero pseudo-proper decay time and those following an exponential decay distribution (both convolved with a detector resolution function) statistically distinguish prompt and non-prompt signal processes, respectively.\nVarious sources of background processes include Drell-Yan processes, mis-reconstructed muon pairs from prompt and non-prompt sources,\nand semileptonic decays from separate $b$-hadrons. \n\nThe probability density function (PDF) for each fit is defined as a normalized sum, \nwhere each term represents a specific signal or background contribution, with a physically motivated mass and $\\tau$ dependence.\nThe PDF can be written in a compact form as\n\\begin{equation}\n\\label{eqn:pdf}\n\\mathrm{PDF}(m,\\tau) = \\sum_{i=1}^{7} \\kappa_i f_i(m) \\ \\cdot h_i(\\tau) \\otimes R(\\tau),\n\\end{equation}\nwhere $\\kappa_i$ represents the relative normalization of the $i^\\mathrm{th}$ term of the seven considered signal and background contributions (such that $\\sum_i \\kappa_i = 1$), \n$f_i(m)$ is the mass-dependent term, and $\\otimes$ represents the convolution of \nthe $\\tau$-dependent function $h_i(\\tau)$ with the $\\tau$ resolution term, $R(\\tau)$. \nThe latter is modelled by a double Gaussian distribution with both means fixed to zero and widths determined from the fit.\n\nTable \\ref{table:fitModel} lists the contributions to the overall PDF with the corresponding $f_i$ and\n$h_i$ functions. Here $G_1$ and $G_2$ are Gaussian functions, $B_1$ and $B_2$ are Crystal Ball\\footnote{The Crystal Ball function is given by:\n\\\\$B(x;\\alpha,n,\\bar x,\\sigma) = N \\cdot \\begin{cases} \\exp\\left(- \\frac{(x - \\bar x)^2}{2 \\sigma^2}\\right), & \\mbox{for }\\frac{x - \\bar x}{\\sigma} > -\\alpha \\\\\n A \\cdot \\left(A' - \\frac{x - \\bar x}{\\sigma}\\right)^{-n}, & \\mbox{for }\\frac{x - \\bar x}{\\sigma} \\leqslant -\\alpha \\end{cases}$ \\\\where $A = \\left(\\frac{n}{\\left| \\alpha \\right|}\\right)^n \\cdot \\exp\\left(- \\frac {\\left| \\alpha \\right|^2}{2}\\right),\nA' = \\frac{n}{\\left| \\alpha \\right|} - \\left| \\alpha \\right|$}\ndistributions~\\cite{CB1}, \nwhile F is a uniform distribution and $C_1$ a first-order Chebyshev polynomial. \nThe exponential functions $E_1$, $E_2$, $E_3$, $E_4$ and $E_5$\nhave different decay constants, where $E_5(|\\tau|)$ is a double-sided exponential with the same decay constant on\neither side of $\\tau = 0$. The parameter $\\omega$ represents the fractional contribution \nof the $B$ and $G$ mass signal functions, while the Dirac delta function, $\\delta(\\tau)$,\nis used to represent the pseudo-proper decay time distribution of the prompt candidates.\n\n\\begin{table}[h!]\n \\centering\n \\caption{Description of the fit model PDF in Eq.~\\ref{eqn:pdf}. Components of the probability density function used to extract the prompt (P) and\nnon-prompt (NP) contributions for \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ signal and the P, NP, and incoherent or mis-reconstructed background (Bkg) contributions.}\n \\begin{tabular}{ l c c c c }\n \\hline \\hline\n $i$ & Type & Source & $f_i(m)$ & $h_i(\\tau)$ \\\\ \\hline \\hline\n 1 & \\jpsi\\ & P & $\\omega B_1(m) + (1-\\omega) G_1(m)$ & $\\delta (\\tau)$ \\\\\n 2 & \\jpsi\\ & NP & $\\omega B_1(m) + (1-\\omega) G_1(m)$ & $ E_1(\\tau)$ \\\\\n 3 & \\ensuremath{\\psi(2\\mathrm{S})}\\ & P & $\\omega B_2(m) + (1-\\omega) G_2(m)$ & $\\delta (\\tau)$ \\\\\n 4 & \\ensuremath{\\psi(2\\mathrm{S})}\\ & NP & $\\omega B_2(m) + (1-\\omega) G_2(m)$ & $E_2(\\tau)$ \\\\ \\hline\n 5 & Bkg & P & $ F$ & $\\delta (\\tau)$ \\\\\n 6 & Bkg & NP & $ C_1(m)$ & $E_3 (\\tau)$ \\\\ \n 7 & Bkg & NP & $ E_4(m)$ & $E_5 (|\\tau|)$ \\\\\n \\hline \n \\end{tabular}\n \\label{table:fitModel}\n\\end{table}\n\nIn order to make the fitting procedure more robust and to reduce the number of free parameters, a number of component terms share common parameters, \nwhich led to 22 free parameters per interval.\nIn detail, the signal mass models are described by the sum of a Crystal Ball shape ($B$) and a Gaussian shape ($G$). For each\nof \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}, the $B$ and $G$ share a common mean, and freely determined widths, \nwith the ratio of the $B$ and $G$ widths common to \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}.\nThe $B$ parameters $\\alpha$, and $n$, describing the transition point of the low-edge from a Gaussian to a power-law \nshape, and the shape of the tail, respectively, are fixed, and variations are considered as part of the fit model\nsystematic uncertainties.\nThe width of $G$ for \\ensuremath{\\psi(2\\mathrm{S})}\\ is set to the width for \\jpsi\\ multiplied by a free parameter scaling term.\nThe relative fraction of $B$ and $G$ is left floating, but common to \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}.\n\nThe non-prompt signal decay shapes ($E_1$,$E_2$) are described by an exponential function (for positive $\\tau$ only) convolved with a\ndouble Gaussian function, $R(\\tau)$ describing the pseudo-proper decay time resolution for the non-prompt component, \nand the same Gaussian response functions to\ndescribe the prompt contributions. \nEach Gaussian resolution component has its mean fixed at $\\tau$ = 0 and a free width.\nThe decay constants of the \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ are separate free parameters in the fit.\n\nThe background contributions are described by a prompt and non-prompt component, as well as a double-sided exponential function\nconvolved with a double Gaussian function describing mis-reconstructed or non-coherent muon pairs. \nThe same resolution function as in signal is used to describe the background. For the non-resonant mass\nparameterizations, the non-prompt contribution is modelled by a first-order Chebyshev polynomial. The prompt mass\ncontribution follows a flat distribution and the double-sided background uses an exponential function.\nVariations of this fit model are considered as systematic uncertainties.\n\nThe following quantities are extracted directly from the fit in each interval:\nthe fraction of events that are signal (prompt or non-prompt \\jpsi\\ or \\ensuremath{\\psi(2\\mathrm{S})}); the fraction of signal events that are prompt;\nthe fraction of prompt signal that is \\ensuremath{\\psi(2\\mathrm{S})}; and the fraction of non-prompt signal that is \\ensuremath{\\psi(2\\mathrm{S})}. From these\nparameters, and the weighted sum of events, all measured values are calculated.\n\n\nFor $7$~\\TeV\\ data, 168 fits are performed across the range of $8<\\mbox{$p_{\\text{T}}$}<100$~\\GeV\\ ($8<\\mbox{$p_{\\text{T}}$}<60$~\\GeV) for $\\jpsi$ ($\\ensuremath{\\psi(2\\mathrm{S})}$) and $0<|y|<2$.\nFor $8$~\\TeV\\ data, 172 fits are performed across the range of $8<\\mbox{$p_{\\text{T}}$}<110$~\\GeV\\ and $0<|y|<2$,\nexcluding the area where $\\mbox{$p_{\\text{T}}$}$ is less than 10 \\GeV\\ and simultaneously $|y|$ is\ngreater than 0.75. This region is excluded due to a steeply changing low trigger efficiency \ncausing large systematic uncertainties in the measured cross-section.\n\nFigure~\\ref{fig:fitprojmain} shows the fit results for one of the intervals considered in the analysis, projected onto the invariant mass and pseudo-proper decay time distributions, for $7$~\\TeV\\ data, weighted according to the acceptance and efficiency corrections.\nThe fit projections are shown for the total prompt and total non-prompt contributions (shown as curves), and also for the individual contributions of the $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$ prompt and non-prompt signal yields (shown as hashed areas of various types). \n\nIn Figure~\\ref{fig:fitprojmainVIII} the fit results are shown for one high-$\\mbox{$p_{\\text{T}}$}$ interval of $8$~\\TeV\\ data.\n\n\\begin{figure} [!ht]\n \\begin{center} \n \\includegraphics[width=0.51\\textwidth]{figures\/c_c_Mass_5_5.eps}\n \\hspace{-0.55cm}\n \\includegraphics[width=0.51\\textwidth]{figures\/c_c_Tau_5_5.eps}\n \\caption{Projections of the fit result over the mass (left) and pseudo-proper decay time (right) distributions for data collected at $7$~\\TeV\\ for one typical interval. The data are shown with error bars in black, superimposed with the individual components of the fit result projections, where the total prompt and non-prompt components are represented by the dashed and dotted lines, respectively, and the shaded areas show the signal $\\psi$ prompt and non-prompt contributions.\\label{fig:fitprojmain}}\n \\end{center}\n\\end{figure} \n\n\\begin{figure} [!ht]\n \\begin{center} \n \\includegraphics[width=0.51\\textwidth]{figures\/c_8TeV_c_Mass_23_0.eps}\n \\hspace{-0.55cm}\n \\includegraphics[width=0.51\\textwidth]{figures\/c_8TeV_c_Tau_23_0.eps}\n \\caption{Projections of the fit result over the mass (left) and pseudo-proper decay time (right) distributions for data collected at $8$~\\TeV\\ for one high-$\\mbox{$p_{\\text{T}}$}$ interval. The data are shown with error bars in black, superimposed with the individual components of the fit result projections, where the total prompt and non-prompt components are represented by the dashed and dotted lines, respectively, and the shaded areas show the signal $\\psi$ prompt and non-prompt contributions.\\label{fig:fitprojmainVIII}}\n \\end{center}\n\\end{figure} \n\n\n\\subsection{Bin migration corrections}\nTo account for bin migration effects due to the detector resolution, \nwhich results in decays of~$\\ensuremath{\\psi}$ in one bin, being identified and accounted for in another,\nthe numbers of acceptance- and efficiency-corrected dimuon decays extracted from\nthe fits in each interval of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ and rapidity are corrected for the differences\nbetween the true and reconstructed values of the dimuon $\\mbox{$p_{\\text{T}}$}$.\nThese corrections are derived from data by comparing analytic functions that are \nfitted to the $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ spectra of dimuon events with and without\nconvolution by the experimental resolution in $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ (as determined from the fitted mass resolution and measured muon angular resolutions), as described in Ref.~\\cite{Aad2012dlq}.\n\nThe correction factors applied to the fitted yields deviate from unity by no more than $1.5\\%$, and for the majority of slices are smaller than $1\\%$.\nThe ratio measurement and non-prompt fractions are corrected by the corresponding ratios of\nbin migration correction factors.\nUsing a similar technique, bin migration corrections as a function of $|y|$ are found \nto differ from unity by negligible amounts.\n\n\n\n\\section*{Acknowledgements}\n\\input{Acknowledgements}\n\n\\clearpage\n\\printbibliography\n\\clearpage\n\n\\section{Results}\n\\label{sec:results}\n\n\nThe $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$ non-prompt and prompt production cross-sections are presented, corrected for acceptance and detector efficiencies \nwhile assuming isotropic decay, as described in Section~\\ref{sec:s:diff_xSecDet}. \nAlso presented are the ratios of non-prompt production relative to the inclusive production for $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$ mesons separately, described in Section~\\ref{sec:s:NPFDet},\nand the ratio of $\\ensuremath{\\psi(2\\mathrm{S})}$ to $\\jpsi$ production for prompt and non-prompt components separately, described in Section~\\ref{sec:s:PNPRatioDet}.\nCorrection factors for various spin-alignment hypotheses for both 7 and 8~\\TeV\\ data can be found in \nTables~\\ref{tab:sa_long_jpsi}--\\ref{tab:sa_offN_psi2s} (in Appendix) and Tables~\\ref{tab:sa_long_jpsi8}--\\ref{tab:sa_offN_psi2s8} (in Appendix) respectively, in terms of \n\\mbox{$p_{\\text{T}}$}\\ and rapidity intervals.\n\n\\setdescription{font=\\normalfont\\itshape}\n\n\\begin{description}[style=unboxed,leftmargin=0cm]\n\n\n\\item[Production cross-sections] \\hfill\n\nFigures \\ref{fig:res:xSecP} and \\ref{fig:res:xSecNP} show respectively the prompt and non-prompt \ndifferential cross-sections of \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ as functions of $\\mbox{$p_{\\text{T}}$}$ and $|y|$, together with the relevant theoretical predictions, \nwhich are described below.\n\n\n\\begin{figure} [!ht]\n \\begin{center}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_JpsiP_xSec.eps} \n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_Psi2SP_xSec.eps}\\hfil\\\\\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_JpsiP_xSec.eps}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_Psi2SP_xSec.eps}\\hfil\n \\caption{The differential prompt cross-section times dimuon branching fraction of \\jpsi\\ (left) and \\ensuremath{\\psi(2\\mathrm{S})}\\ (right) as a function \n of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ for each slice of rapidity. \n The top (bottom) row shows the 7~\\TeV\\ (8~\\TeV) results.\n For each increasing rapidity slice, an additional scaling factor of 10 is applied to the plotted points for visual clarity. The\n centre of each bin on the horizontal axis represents the mean of the weighted $\\mbox{$p_{\\text{T}}$}$ distribution. The\n horizontal error bars represent the range of $\\mbox{$p_{\\text{T}}$}$ for the bin, and the vertical error bar covers \n the statistical and systematic \n uncertainty (with the same multiplicative scaling applied). \n The NLO NRQCD theory predictions are also shown.}\n \\label{fig:res:xSecP}\n \\end{center}\n\\end{figure} \n\n\\begin{figure} [!ht]\n \\begin{center} \n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_JpsiNP_xSec.eps}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_Psi2SNP_xSec.eps}\\hfil\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_JpsiNP_xSec.eps}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_Psi2SNP_xSec.eps}\\hfil\n \\caption{The differential non-prompt cross-section times dimuon branching fraction of \\jpsi\\ (left) and \\ensuremath{\\psi(2\\mathrm{S})}\\ (right) as a function \n of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ for each slice of rapidity. \n The top (bottom) row shows the 7~\\TeV\\ (8~\\TeV) results.\n For each increasing rapidity slice, an additional scaling factor of 10 is applied to the plotted points for visual clarity. The\n centre of each bin on the horizontal axis represents the mean of the weighted $\\mbox{$p_{\\text{T}}$}$ distribution. The\n horizontal error bars represent the range of $\\mbox{$p_{\\text{T}}$}$ for the bin, and the vertical error bar covers the statistical\n and systematic uncertainty (with the same multiplicative scaling applied).\n The FONLL theory predictions are also shown.}\n \\label{fig:res:xSecNP}\n \\end{center}\n\\end{figure} \n\n\n\\item[Non-prompt production fractions] \\hfill\n\nThe results for the fractions of non-prompt production relative to the inclusive production of \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\, are presented as a function of $\\mbox{$p_{\\text{T}}$}$ for slices of rapidity in Figure~\\ref{fig:res:NPF}. \nIn each rapidity slice, the non-prompt fraction is seen to increase as a function of $\\mbox{$p_{\\text{T}}$}$ and has no strong dependence on either rapidity or centre-of-mass energy.\n\n\\begin{figure} [!ht]\n \\begin{center}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_NPF_Jpsi.eps} \n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_NPF_Psi.eps}\\hfil\\\\\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_NPF_Jpsi.eps}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_NPF_Psi.eps}\\hfil\n \\caption{The non-prompt fraction of \\jpsi\\ (left) and \\ensuremath{\\psi(2\\mathrm{S})}\\ (right), as a function of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ for each slice of rapidity. \n The top (bottom) row shows the 7~\\TeV\\ (8~\\TeV) results.\n For each increasing rapidity slice, an additional factor of 0.2 is applied to the plotted points for visual clarity. The\n centre of each bin on the horizontal axis represents the mean of the weighted $\\mbox{$p_{\\text{T}}$}$ distribution. The\n horizontal error bars represent the range of $\\mbox{$p_{\\text{T}}$}$ for the bin, and the vertical error bar covers the statistical\n and systematic uncertainty (with the same multiplicative scaling applied).}\n \\label{fig:res:NPF}\n \\end{center}\n\\end{figure} \n\n\n\\textit{Production ratios of \\textmd{\\ensuremath{\\psi(2\\mathrm{S})}} to \\textmd{\\jpsi} }\n\nFigure~\\ref{fig:res:PNP_Ratio} shows the ratios of \\ensuremath{\\psi(2\\mathrm{S})}\\ to \\jpsi\\ decaying to a muon pair in prompt and non-prompt processes,\n presented as a function of $\\mbox{$p_{\\text{T}}$}$ for slices of rapidity. The non-prompt ratio is shown to be relatively flat across the considered range of $\\pT$,\nfor each slice of rapidity.\nFor the prompt ratio, a slight increase as a function of $\\mbox{$p_{\\text{T}}$}$ is observed, with no strong dependence on rapidity or centre-of-mass energy.\n\n\\begin{figure} [!ht]\n \\begin{center}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_P_Ratio.eps} \n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_NP_Ratio.eps}\\hfil\\\\\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_P_Ratio.eps}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_NP_Ratio.eps}\\hfil\n \\caption{The ratio of \\ensuremath{\\psi(2\\mathrm{S})}\\ to \\jpsi\\ production times dimuon branching fraction for prompt (left) and non-prompt (right) processes\n as a function of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ for each of the slices of rapidity. \n For each increasing rapidity slice, an additional factor of 0.1 is applied to the plotted points for visual clarity.\n The top (bottom) row shows the 7~\\TeV\\ (8~\\TeV) results.\n The centre of each bin on the horizontal axis represents the mean of the weighted $\\mbox{$p_{\\text{T}}$}$ distribution. The\n horizontal error bars represent the range of $\\mbox{$p_{\\text{T}}$}$ for the bin, and the vertical error bar covers the statistical and systematic uncertainty.}\n \\label{fig:res:PNP_Ratio}\n \\end{center}\n\\end{figure} \n\n\\clearpage\n\n\n\\item[Comparison with theory] \\hfill\n\nFor prompt production, as shown in Figure~\\ref{fig:xsecPtheoryRatio}, the ratio of the NLO NRQCD theory calculations~\\cite{NRQCD1} \nto data, as a function of~$\\mbox{$p_{\\text{T}}$}$ and in slices of rapidity, is provided for \\Jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ at both the 7 and 8~\\TeV\\ centre-of-mass energies.\nThe theory predictions are \nbased on the long-distance matrix elements (LDMEs) from Refs.~\\cite{NRQCD1,Ma:2010vd}, with uncertainties originating from the\nchoice of scale, charm quark mass and LDMEs (see Refs.~\\cite{NRQCD1,Ma:2010vd} for more details).\nFigure~\\ref{fig:xsecPtheoryRatio} shows fair agreement between the theoretical calculation and the data points for the whole $\\mbox{$p_{\\text{T}}$}$ range. \nThe ratio of theory to data does not depend on rapidity.\n\n\\begin{figure} [!ht]\n \\begin{center}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_JpsiP_theoryRatio_lin.eps} \n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_Psi2SP_theoryRatio_lin.eps}\\hfil\\\\\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_JpsiP_theoryRatio_lin.eps}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_Psi2SP_theoryRatio_lin.eps}\\hfil\n \\caption{The ratios of the NRQCD theoretical predictions to data are presented for the differential prompt cross-section of \\jpsi\\ (left) and \\ensuremath{\\psi(2\\mathrm{S})}\\ (right) as a function \n of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ for each rapidity slice. \n The top (bottom) row shows the 7~\\TeV\\ (8~\\TeV) results.\n The error on the data is the relative error of each data point, while the error bars on the theory prediction are the relative error of each theory point.}\n \\label{fig:xsecPtheoryRatio}\n \\end{center}\n\\end{figure} \n\nFor non-prompt $\\psi$ production, comparisons are made to FONLL theoretical predictions~\\cite{FONLL_2001,Cacciari:2012ny}, which describe the production of \n$b$-hadrons followed by their decay into $\\psi+X$.\nFigure~\\ref{fig:xsecNPtheoryRatio} shows the ratios of $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$ FONLL predictions to data, as a function of $\\mbox{$p_{\\text{T}}$}$ and in slices of rapidity, \nfor centre-of-mass energies of~7 and 8~\\TeV.\nFor $\\jpsi$, agreement is generally good, but the theory predicts slightly harder $\\mbox{$p_{\\text{T}}$}$ spectra than observed in the data.\nFor $\\ensuremath{\\psi(2\\mathrm{S})}$, the shapes of data and theory appear to be in satisfactory agreement, but the theory predicts higher yields than in the data.\nThere is no observed dependence on rapidity in the comparisons between theory and data for non-prompt \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ production.\n\n\n\\begin{figure} [!ht]\n \\begin{center}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_JpsiNP_theoryRatio_lin.eps} \n \\includegraphics[width=0.44\\textwidth]{figures\/ct_7TeV_Psi2SNP_theoryRatio_lin.eps}\\hfil\\\\\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_JpsiNP_theoryRatio_lin.eps}\n \\includegraphics[width=0.44\\textwidth]{figures\/ct_8TeV_Psi2SNP_theoryRatio_lin.eps}\\hfil\n \\caption{The ratio of the FONLL theoretical predictions to data are presented for the differential non-prompt \n cross-section of \\jpsi\\ (left) and \\ensuremath{\\psi(2\\mathrm{S})}\\ (right) as a function \n of $\\mbox{$p_{\\text{T}}$}(\\mu\\mu)$ for each rapidity slice. \n The top (bottom) row shows the 7~\\TeV\\ (8~\\TeV) results.\n The error on the data is the relative error of each data point, while the error bars on the theory prediction are the relative error of each theory point.}\n \\label{fig:xsecNPtheoryRatio}\n \\end{center}\n\\end{figure} \n\n\n\\item[Comparison of cross-sections 8~\\TeV\\ with 7~\\TeV\\ ] \\hfill\n\nIt is interesting to compare the cross-section results between the two centre-of-mass energies, \nboth for data and the theoretical predictions.\n\nFigure~\\ref{fig:xsecEnergyRatio} shows the 8~\\TeV\\ to 7~\\TeV\\ cross-section ratios of prompt and non-prompt $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$ for both data sets.\nFor the theoretical ratios the uncertainties are neglected here, since the high correlation between them results in large cancellations.\n\n\nDue to a finer granularity in $\\mbox{$p_{\\text{T}}$}$ for the 8~\\TeV\\ data, a weighted average of the 8~\\TeV\\ results is taken across equivalent intervals of the 7~\\TeV\\ data to enable direct comparisons.\nBoth data and theoretical predictions agree \nthat the ratios become larger with increasing $\\mbox{$p_{\\text{T}}$}$,\nhowever at the lower edge of the $\\pT$ range the data tends to be slightly below theory.\n\n\n\\begin{figure} [!ht]\n \\begin{center}\n \\includegraphics[width=0.44\\textwidth]{figures\/rt_th_xSecP_J.eps} \n \\includegraphics[width=0.44\\textwidth]{figures\/rt_th_xSecP_P.eps}\\hfil\\\\\n \\includegraphics[width=0.44\\textwidth]{figures\/rt_th_xSecNP_J.eps}\n \\includegraphics[width=0.44\\textwidth]{figures\/rt_th_xSecNP_P.eps}\\hfil\n \\caption{The ratio of the 8~\\TeV\\ and 7~\\TeV\\ differential cross-sections\n\t are presented for prompt~(top) and non-prompt~(bottom) $\\jpsi$~(left) and \n\t $\\ensuremath{\\psi(2\\mathrm{S})}$~(right) for both data~(red points with error bars) and theoretical predictions~(green points).\n\t The theoretical predictions used are NRQCD for prompt and FONLL for non-prompt production.\n\t The uncertainty on the data ratio does not account for possible correlations between 7 and 8 \\TeV\\ data, and no uncertainty is shown \n\t for the ratio of theory predictions.\n}\n \\label{fig:xsecEnergyRatio}\n \\end{center}\n\\end{figure} \n\n\n\\end{description}\n\n\\section{Spin-alignment correction factors}\n\\label{sec:spincorrection}\n\nThe measurement presented here assumes an unpolarized spin-alignment hypothesis for determining the correction factor. In principle, the polarization may be non-zero and may vary with $\\mbox{$p_{\\text{T}}$}$.\nIn order to correct these measurements when well-measured $\\jpsi$ and $\\ensuremath{\\psi(2\\mathrm{S})}$ polarizations are determined, a set of correction factors are provided \nin Tables~\\ref{tab:sa_long_jpsi}--\\ref{tab:sa_offN_psi2s} for the 7~\\TeV\\ data, and in the Tables~\\ref{tab:sa_long_jpsi8}--\\ref{tab:sa_offN_psi2s8} for the 8~\\TeV\\ data.\nThese tables are created by altering the spin-alignment hypothesis for either the $\\jpsi$ or $\\ensuremath{\\psi(2\\mathrm{S})}$ meson and then determining the ratio of the mean sum-of-weights of the new hypotheses to the original flat hypothesis.\nThe mean weight is calculated from all the events in each dimuon $\\mbox{$p_{\\text{T}}$}$ and rapidity analysis bin, selecting those dimuons within $\\pm2\\sigma$ of the $\\ensuremath{\\psi}$ fitted mean mass position.\nThe choice of spin-alignment hypothesis for each $\\ensuremath{\\psi}$ meson has negligible effect on the results of the other $\\ensuremath{\\psi}$ meson, and therefore these possible permutations are not considered.\nThe definitions of each of the spin-alignment scenarios, which are given in the caption to the table, are defined in Table~\\ref{tab:spin}.\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``longitudinal'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 0.666 & 0.672 & 0.674 & 0.680 & 0.688 & 0.690 & 0.690 & 0.690 \\\\\n$8.50$--$9.00$ & 0.670 & 0.674 & 0.678 & 0.685 & 0.689 & 0.694 & 0.694 & 0.698 \\\\\n$9.00$--$9.50$ & 0.673 & 0.676 & 0.680 & 0.687 & 0.693 & 0.697 & 0.698 & 0.700 \\\\\n$9.50$--$10.00$ & 0.675 & 0.678 & 0.683 & 0.689 & 0.694 & 0.697 & 0.701 & 0.703 \\\\\n$10.00$--$10.50$ & 0.679 & 0.681 & 0.687 & 0.692 & 0.697 & 0.699 & 0.702 & 0.706 \\\\\n$10.50$--$11.00$ & 0.682 & 0.686 & 0.691 & 0.696 & 0.700 & 0.702 & 0.704 & 0.708 \\\\\n$11.00$--$11.50$ & 0.688 & 0.689 & 0.694 & 0.699 & 0.701 & 0.705 & 0.708 & 0.710 \\\\\n$11.50$--$12.00$ & 0.692 & 0.695 & 0.698 & 0.702 & 0.706 & 0.708 & 0.710 & 0.712 \\\\\n$12.00$--$13.00$ & 0.698 & 0.700 & 0.703 & 0.707 & 0.711 & 0.713 & 0.715 & 0.717 \\\\\n$13.00$--$14.00$ & 0.707 & 0.709 & 0.711 & 0.714 & 0.717 & 0.720 & 0.721 & 0.723 \\\\\n$14.00$--$15.00$ & 0.716 & 0.717 & 0.720 & 0.722 & 0.725 & 0.727 & 0.728 & 0.730 \\\\\n$15.00$--$16.00$ & 0.724 & 0.726 & 0.728 & 0.729 & 0.732 & 0.734 & 0.735 & 0.737 \\\\\n$16.00$--$17.00$ & 0.733 & 0.733 & 0.735 & 0.737 & 0.739 & 0.741 & 0.742 & 0.744 \\\\\n$17.00$--$18.00$ & 0.740 & 0.741 & 0.743 & 0.744 & 0.746 & 0.747 & 0.749 & 0.750 \\\\\n$18.00$--$20.00$ & 0.751 & 0.752 & 0.753 & 0.754 & 0.756 & 0.758 & 0.758 & 0.760 \\\\\n$20.00$--$22.00$ & 0.765 & 0.765 & 0.766 & 0.767 & 0.769 & 0.770 & 0.771 & 0.772 \\\\\n$22.00$--$24.00$ & 0.777 & 0.777 & 0.778 & 0.780 & 0.781 & 0.781 & 0.782 & 0.783 \\\\\n$24.00$--$26.00$ & 0.789 & 0.789 & 0.790 & 0.790 & 0.791 & 0.792 & 0.793 & 0.794 \\\\\n$26.00$--$30.00$ & 0.803 & 0.803 & 0.804 & 0.804 & 0.805 & 0.806 & 0.806 & 0.807 \\\\\n$30.00$--$40.00$ & 0.827 & 0.827 & 0.828 & 0.828 & 0.829 & 0.829 & 0.830 & 0.831 \\\\\n$40.00$--$60.00$ & 0.863 & 0.863 & 0.864 & 0.864 & 0.864 & 0.865 & 0.865 & 0.866 \\\\\n$60.00$--$100.00$ & 0.902 & 0.904 & 0.904 & 0.903 & 0.904 & 0.904 & 0.902 & 0.906 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_long_jpsi} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``transverse zero'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 1.336 & 1.324 & 1.315 & 1.309 & 1.299 & 1.297 & 1.296 & 1.298 \\\\\n$8.50$--$9.00$ & 1.329 & 1.323 & 1.310 & 1.300 & 1.291 & 1.284 & 1.280 & 1.284 \\\\\n$9.00$--$9.50$ & 1.326 & 1.315 & 1.303 & 1.295 & 1.289 & 1.281 & 1.279 & 1.276 \\\\\n$9.50$--$10.00$ & 1.317 & 1.311 & 1.300 & 1.289 & 1.284 & 1.276 & 1.276 & 1.272 \\\\\n$10.00$--$10.50$ & 1.310 & 1.304 & 1.297 & 1.290 & 1.280 & 1.276 & 1.273 & 1.269 \\\\\n$10.50$--$11.00$ & 1.302 & 1.298 & 1.291 & 1.285 & 1.276 & 1.271 & 1.267 & 1.268 \\\\\n$11.00$--$11.50$ & 1.296 & 1.290 & 1.284 & 1.278 & 1.271 & 1.266 & 1.263 & 1.261 \\\\\n$11.50$--$12.00$ & 1.288 & 1.284 & 1.277 & 1.274 & 1.265 & 1.261 & 1.260 & 1.257 \\\\\n$12.00$--$13.00$ & 1.276 & 1.273 & 1.268 & 1.263 & 1.257 & 1.255 & 1.251 & 1.250 \\\\\n$13.00$--$14.00$ & 1.263 & 1.260 & 1.254 & 1.250 & 1.247 & 1.244 & 1.243 & 1.240 \\\\\n$14.00$--$15.00$ & 1.248 & 1.246 & 1.244 & 1.240 & 1.236 & 1.233 & 1.233 & 1.230 \\\\\n$15.00$--$16.00$ & 1.237 & 1.233 & 1.231 & 1.228 & 1.225 & 1.223 & 1.223 & 1.221 \\\\\n$16.00$--$17.00$ & 1.224 & 1.222 & 1.221 & 1.219 & 1.216 & 1.213 & 1.212 & 1.212 \\\\\n$17.00$--$18.00$ & 1.213 & 1.213 & 1.211 & 1.208 & 1.205 & 1.204 & 1.204 & 1.203 \\\\\n$18.00$--$20.00$ & 1.200 & 1.198 & 1.197 & 1.196 & 1.194 & 1.192 & 1.192 & 1.190 \\\\\n$20.00$--$22.00$ & 1.183 & 1.182 & 1.180 & 1.180 & 1.178 & 1.177 & 1.176 & 1.175 \\\\\n$22.00$--$24.00$ & 1.168 & 1.167 & 1.166 & 1.165 & 1.164 & 1.164 & 1.163 & 1.163 \\\\\n$24.00$--$26.00$ & 1.155 & 1.155 & 1.154 & 1.154 & 1.153 & 1.152 & 1.151 & 1.150 \\\\\n$26.00$--$30.00$ & 1.140 & 1.140 & 1.140 & 1.139 & 1.138 & 1.138 & 1.138 & 1.137 \\\\\n$30.00$--$40.00$ & 1.117 & 1.117 & 1.117 & 1.116 & 1.116 & 1.116 & 1.115 & 1.115 \\\\\n$40.00$--$60.00$ & 1.087 & 1.087 & 1.086 & 1.086 & 1.086 & 1.085 & 1.085 & 1.085 \\\\\n$60.00$--$100.00$ & 1.057 & 1.056 & 1.057 & 1.057 & 1.056 & 1.056 & 1.057 & 1.055 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trp0_jpsi} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``transverse positive'' spin-alignment transverse positive hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 1.693 & 1.694 & 1.700 & 1.711 & 1.727 & 1.720 & 1.720 & 1.747 \\\\\n$8.50$--$9.00$ & 1.561 & 1.564 & 1.564 & 1.568 & 1.568 & 1.568 & 1.571 & 1.673 \\\\\n$9.00$--$9.50$ & 1.468 & 1.468 & 1.465 & 1.466 & 1.470 & 1.466 & 1.471 & 1.519 \\\\\n$9.50$--$10.00$ & 1.418 & 1.416 & 1.417 & 1.417 & 1.421 & 1.417 & 1.423 & 1.453 \\\\\n$10.00$--$10.50$ & 1.383 & 1.383 & 1.387 & 1.389 & 1.390 & 1.389 & 1.391 & 1.406 \\\\\n$10.50$--$11.00$ & 1.360 & 1.362 & 1.365 & 1.364 & 1.364 & 1.362 & 1.362 & 1.380 \\\\\n$11.00$--$11.50$ & 1.344 & 1.342 & 1.342 & 1.344 & 1.344 & 1.344 & 1.346 & 1.355 \\\\\n$11.50$--$12.00$ & 1.326 & 1.326 & 1.327 & 1.329 & 1.327 & 1.327 & 1.329 & 1.334 \\\\\n$12.00$--$13.00$ & 1.307 & 1.308 & 1.307 & 1.308 & 1.308 & 1.308 & 1.308 & 1.312 \\\\\n$13.00$--$14.00$ & 1.285 & 1.287 & 1.285 & 1.285 & 1.285 & 1.286 & 1.285 & 1.288 \\\\\n$14.00$--$15.00$ & 1.266 & 1.266 & 1.267 & 1.267 & 1.266 & 1.266 & 1.266 & 1.268 \\\\\n$15.00$--$16.00$ & 1.250 & 1.249 & 1.250 & 1.251 & 1.250 & 1.249 & 1.250 & 1.250 \\\\\n$16.00$--$17.00$ & 1.234 & 1.235 & 1.235 & 1.235 & 1.235 & 1.235 & 1.235 & 1.235 \\\\\n$17.00$--$18.00$ & 1.222 & 1.223 & 1.223 & 1.223 & 1.222 & 1.222 & 1.223 & 1.222 \\\\\n$18.00$--$20.00$ & 1.206 & 1.206 & 1.206 & 1.207 & 1.207 & 1.207 & 1.206 & 1.205 \\\\\n$20.00$--$22.00$ & 1.187 & 1.187 & 1.187 & 1.188 & 1.187 & 1.187 & 1.187 & 1.186 \\\\\n$22.00$--$24.00$ & 1.171 & 1.171 & 1.171 & 1.171 & 1.171 & 1.171 & 1.171 & 1.171 \\\\\n$24.00$--$26.00$ & 1.158 & 1.158 & 1.158 & 1.158 & 1.158 & 1.158 & 1.158 & 1.156 \\\\\n$26.00$--$30.00$ & 1.142 & 1.142 & 1.142 & 1.142 & 1.142 & 1.142 & 1.142 & 1.141 \\\\\n$30.00$--$40.00$ & 1.118 & 1.118 & 1.118 & 1.118 & 1.118 & 1.118 & 1.118 & 1.117 \\\\\n$40.00$--$60.00$ & 1.087 & 1.087 & 1.087 & 1.086 & 1.087 & 1.086 & 1.086 & 1.086 \\\\\n$60.00$--$100.00$ & 1.058 & 1.057 & 1.057 & 1.057 & 1.057 & 1.056 & 1.058 & 1.056 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trpp_jpsi} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``transverse negative'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 1.030 & 1.020 & 1.004 & 0.995 & 0.992 & 0.981 & 0.973 & 0.949 \\\\\n$8.50$--$9.00$ & 1.157 & 1.148 & 1.134 & 1.113 & 1.101 & 1.092 & 1.079 & 1.055 \\\\\n$9.00$--$9.50$ & 1.207 & 1.196 & 1.176 & 1.161 & 1.147 & 1.138 & 1.130 & 1.107 \\\\\n$9.50$--$10.00$ & 1.231 & 1.219 & 1.202 & 1.186 & 1.174 & 1.162 & 1.158 & 1.138 \\\\\n$10.00$--$10.50$ & 1.243 & 1.231 & 1.217 & 1.202 & 1.189 & 1.181 & 1.175 & 1.158 \\\\\n$10.50$--$11.00$ & 1.246 & 1.239 & 1.228 & 1.213 & 1.200 & 1.191 & 1.186 & 1.174 \\\\\n$11.00$--$11.50$ & 1.252 & 1.242 & 1.230 & 1.218 & 1.205 & 1.198 & 1.192 & 1.181 \\\\\n$11.50$--$12.00$ & 1.251 & 1.243 & 1.229 & 1.222 & 1.208 & 1.202 & 1.197 & 1.187 \\\\\n$12.00$--$13.00$ & 1.247 & 1.240 & 1.230 & 1.221 & 1.211 & 1.205 & 1.200 & 1.193 \\\\\n$13.00$--$14.00$ & 1.240 & 1.235 & 1.227 & 1.218 & 1.211 & 1.206 & 1.202 & 1.197 \\\\\n$14.00$--$15.00$ & 1.232 & 1.227 & 1.221 & 1.215 & 1.207 & 1.203 & 1.200 & 1.195 \\\\\n$15.00$--$16.00$ & 1.223 & 1.219 & 1.213 & 1.207 & 1.201 & 1.198 & 1.196 & 1.193 \\\\\n$16.00$--$17.00$ & 1.213 & 1.210 & 1.206 & 1.201 & 1.196 & 1.193 & 1.191 & 1.189 \\\\\n$17.00$--$18.00$ & 1.204 & 1.203 & 1.199 & 1.194 & 1.189 & 1.187 & 1.186 & 1.183 \\\\\n$18.00$--$20.00$ & 1.193 & 1.191 & 1.188 & 1.185 & 1.181 & 1.179 & 1.177 & 1.176 \\\\\n$20.00$--$22.00$ & 1.178 & 1.177 & 1.174 & 1.172 & 1.169 & 1.167 & 1.166 & 1.164 \\\\\n$22.00$--$24.00$ & 1.164 & 1.163 & 1.162 & 1.159 & 1.157 & 1.156 & 1.156 & 1.154 \\\\\n$24.00$--$26.00$ & 1.153 & 1.152 & 1.150 & 1.149 & 1.148 & 1.147 & 1.145 & 1.144 \\\\\n$26.00$--$30.00$ & 1.139 & 1.138 & 1.137 & 1.136 & 1.135 & 1.134 & 1.133 & 1.132 \\\\\n$30.00$--$40.00$ & 1.116 & 1.116 & 1.115 & 1.114 & 1.114 & 1.113 & 1.113 & 1.112 \\\\\n$40.00$--$60.00$ & 1.086 & 1.086 & 1.086 & 1.085 & 1.085 & 1.084 & 1.084 & 1.084 \\\\\n$60.00$--$100.00$ & 1.057 & 1.056 & 1.056 & 1.056 & 1.056 & 1.056 & 1.057 & 1.055 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trpm_jpsi} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``off-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane positive'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 1.015 & 1.047 & 1.073 & 1.094 & 1.113 & 1.120 & 1.124 & 1.122 \\\\\n$8.50$--$9.00$ & 1.020 & 1.058 & 1.087 & 1.110 & 1.125 & 1.134 & 1.142 & 1.144 \\\\\n$9.00$--$9.50$ & 1.019 & 1.056 & 1.084 & 1.107 & 1.127 & 1.138 & 1.144 & 1.145 \\\\\n$9.50$--$10.00$ & 1.017 & 1.053 & 1.081 & 1.105 & 1.122 & 1.129 & 1.140 & 1.142 \\\\\n$10.00$--$10.50$ & 1.017 & 1.049 & 1.077 & 1.100 & 1.115 & 1.125 & 1.132 & 1.136 \\\\\n$10.50$--$11.00$ & 1.014 & 1.048 & 1.075 & 1.095 & 1.109 & 1.118 & 1.124 & 1.130 \\\\\n$11.00$--$11.50$ & 1.015 & 1.044 & 1.069 & 1.088 & 1.103 & 1.112 & 1.117 & 1.122 \\\\\n$11.50$--$12.00$ & 1.014 & 1.043 & 1.066 & 1.083 & 1.096 & 1.105 & 1.112 & 1.115 \\\\\n$12.00$--$13.00$ & 1.012 & 1.038 & 1.060 & 1.076 & 1.089 & 1.097 & 1.101 & 1.105 \\\\\n$13.00$--$14.00$ & 1.012 & 1.035 & 1.053 & 1.068 & 1.079 & 1.087 & 1.090 & 1.093 \\\\\n$14.00$--$15.00$ & 1.010 & 1.031 & 1.048 & 1.061 & 1.070 & 1.076 & 1.080 & 1.083 \\\\\n$15.00$--$16.00$ & 1.010 & 1.028 & 1.043 & 1.054 & 1.063 & 1.068 & 1.072 & 1.074 \\\\\n$16.00$--$17.00$ & 1.009 & 1.025 & 1.039 & 1.049 & 1.056 & 1.062 & 1.065 & 1.067 \\\\\n$17.00$--$18.00$ & 1.009 & 1.023 & 1.035 & 1.044 & 1.051 & 1.055 & 1.059 & 1.060 \\\\\n$18.00$--$20.00$ & 1.007 & 1.019 & 1.030 & 1.039 & 1.045 & 1.049 & 1.051 & 1.053 \\\\\n$20.00$--$22.00$ & 1.007 & 1.016 & 1.025 & 1.032 & 1.038 & 1.041 & 1.043 & 1.044 \\\\\n$22.00$--$24.00$ & 1.005 & 1.014 & 1.021 & 1.027 & 1.031 & 1.035 & 1.036 & 1.037 \\\\\n$24.00$--$26.00$ & 1.005 & 1.012 & 1.018 & 1.024 & 1.027 & 1.030 & 1.032 & 1.032 \\\\\n$26.00$--$30.00$ & 1.004 & 1.010 & 1.015 & 1.019 & 1.023 & 1.025 & 1.026 & 1.026 \\\\\n$30.00$--$40.00$ & 1.003 & 1.007 & 1.011 & 1.014 & 1.016 & 1.017 & 1.018 & 1.019 \\\\\n$40.00$--$60.00$ & 1.002 & 1.004 & 1.006 & 1.008 & 1.009 & 1.010 & 1.010 & 1.010 \\\\\n$60.00$--$100.00$ & 1.001 & 1.002 & 1.003 & 1.003 & 1.004 & 1.004 & 1.005 & 1.005 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_offP_jpsi} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``off-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane negative'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 0.984 & 0.956 & 0.932 & 0.920 & 0.912 & 0.904 & 0.898 & 0.905 \\\\\n$8.50$--$9.00$ & 0.981 & 0.948 & 0.925 & 0.910 & 0.898 & 0.892 & 0.886 & 0.891 \\\\\n$9.00$--$9.50$ & 0.983 & 0.950 & 0.925 & 0.910 & 0.901 & 0.893 & 0.889 & 0.888 \\\\\n$9.50$--$10.00$ & 0.983 & 0.951 & 0.929 & 0.912 & 0.903 & 0.894 & 0.892 & 0.891 \\\\\n$10.00$--$10.50$ & 0.985 & 0.953 & 0.932 & 0.918 & 0.907 & 0.900 & 0.896 & 0.894 \\\\\n$10.50$--$11.00$ & 0.984 & 0.957 & 0.936 & 0.922 & 0.910 & 0.904 & 0.900 & 0.899 \\\\\n$11.00$--$11.50$ & 0.985 & 0.958 & 0.939 & 0.927 & 0.915 & 0.909 & 0.906 & 0.903 \\\\\n$11.50$--$12.00$ & 0.987 & 0.961 & 0.942 & 0.929 & 0.919 & 0.912 & 0.910 & 0.907 \\\\\n$12.00$--$13.00$ & 0.987 & 0.963 & 0.945 & 0.934 & 0.925 & 0.920 & 0.915 & 0.913 \\\\\n$13.00$--$14.00$ & 0.989 & 0.968 & 0.951 & 0.940 & 0.932 & 0.927 & 0.924 & 0.922 \\\\\n$14.00$--$15.00$ & 0.990 & 0.971 & 0.957 & 0.946 & 0.938 & 0.934 & 0.931 & 0.929 \\\\\n$15.00$--$16.00$ & 0.992 & 0.974 & 0.961 & 0.951 & 0.944 & 0.940 & 0.937 & 0.936 \\\\\n$16.00$--$17.00$ & 0.991 & 0.976 & 0.964 & 0.955 & 0.949 & 0.945 & 0.943 & 0.941 \\\\\n$17.00$--$18.00$ & 0.992 & 0.978 & 0.968 & 0.959 & 0.953 & 0.949 & 0.948 & 0.946 \\\\\n$18.00$--$20.00$ & 0.993 & 0.981 & 0.971 & 0.964 & 0.959 & 0.956 & 0.954 & 0.953 \\\\\n$20.00$--$22.00$ & 0.994 & 0.984 & 0.976 & 0.970 & 0.965 & 0.962 & 0.961 & 0.960 \\\\\n$22.00$--$24.00$ & 0.994 & 0.986 & 0.979 & 0.974 & 0.970 & 0.968 & 0.966 & 0.965 \\\\\n$24.00$--$26.00$ & 0.995 & 0.988 & 0.982 & 0.977 & 0.974 & 0.972 & 0.971 & 0.970 \\\\\n$26.00$--$30.00$ & 0.996 & 0.990 & 0.985 & 0.981 & 0.978 & 0.977 & 0.976 & 0.975 \\\\\n$30.00$--$40.00$ & 0.997 & 0.993 & 0.990 & 0.987 & 0.985 & 0.983 & 0.983 & 0.982 \\\\\n$40.00$--$60.00$ & 0.998 & 0.996 & 0.994 & 0.993 & 0.991 & 0.991 & 0.990 & 0.990 \\\\\n$60.00$--$100.00$ & 0.999 & 0.998 & 0.997 & 0.997 & 0.996 & 0.996 & 0.995 & 0.995 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_offN_jpsi} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``longitudinal'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 0.670 & 0.678 & 0.685 & 0.692 & 0.701 & 0.707 & 0.713 & 0.709 \\\\\n$8.50$--$9.00$ & 0.676 & 0.681 & 0.688 & 0.698 & 0.703 & 0.709 & 0.712 & 0.713 \\\\\n$9.00$--$9.50$ & 0.678 & 0.683 & 0.691 & 0.700 & 0.708 & 0.713 & 0.717 & 0.718 \\\\\n$9.50$--$10.00$ & 0.680 & 0.684 & 0.693 & 0.699 & 0.708 & 0.710 & 0.720 & 0.722 \\\\\n$10.00$--$10.50$ & 0.684 & 0.687 & 0.695 & 0.704 & 0.707 & 0.713 & 0.720 & 0.725 \\\\\n$10.50$--$11.00$ & 0.687 & 0.691 & 0.698 & 0.705 & 0.712 & 0.714 & 0.719 & 0.728 \\\\\n$11.00$--$11.50$ & 0.692 & 0.695 & 0.701 & 0.709 & 0.713 & 0.717 & 0.722 & 0.728 \\\\\n$11.50$--$12.00$ & 0.696 & 0.700 & 0.704 & 0.711 & 0.717 & 0.719 & 0.724 & 0.729 \\\\\n$12.00$--$13.00$ & 0.701 & 0.705 & 0.710 & 0.716 & 0.720 & 0.724 & 0.727 & 0.731 \\\\\n$13.00$--$14.00$ & 0.711 & 0.714 & 0.718 & 0.722 & 0.727 & 0.730 & 0.732 & 0.734 \\\\\n$14.00$--$15.00$ & 0.719 & 0.722 & 0.725 & 0.730 & 0.732 & 0.736 & 0.739 & 0.742 \\\\\n$15.00$--$16.00$ & 0.727 & 0.729 & 0.733 & 0.735 & 0.740 & 0.741 & 0.745 & 0.745 \\\\\n$16.00$--$17.00$ & 0.736 & 0.738 & 0.740 & 0.743 & 0.746 & 0.748 & 0.749 & 0.753 \\\\\n$17.00$--$18.00$ & 0.742 & 0.744 & 0.748 & 0.750 & 0.753 & 0.754 & 0.759 & 0.760 \\\\\n$18.00$--$20.00$ & 0.753 & 0.755 & 0.758 & 0.760 & 0.762 & 0.763 & 0.764 & 0.769 \\\\\n$20.00$--$22.00$ & 0.766 & 0.767 & 0.770 & 0.773 & 0.774 & 0.776 & 0.776 & 0.778 \\\\\n$22.00$--$24.00$ & 0.778 & 0.782 & 0.780 & 0.784 & 0.785 & 0.782 & 0.790 & 0.788 \\\\\n$24.00$--$26.00$ & 0.791 & 0.791 & 0.795 & 0.795 & 0.795 & 0.799 & 0.798 & 0.798 \\\\\n$26.00$--$30.00$ & 0.806 & 0.805 & 0.805 & 0.809 & 0.808 & 0.810 & 0.810 & 0.812 \\\\\n$30.00$--$40.00$ & 0.829 & 0.830 & 0.830 & 0.830 & 0.828 & 0.832 & 0.830 & 0.830 \\\\\n$40.00$--$60.00$ & 0.864 & 0.865 & 0.867 & 0.864 & 0.868 & 0.867 & 0.861 & 0.953 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_long_psi2s} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``transverse zero'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 1.328 & 1.311 & 1.300 & 1.284 & 1.274 & 1.267 & 1.261 & 1.265 \\\\\n$8.50$--$9.00$ & 1.318 & 1.309 & 1.293 & 1.279 & 1.268 & 1.263 & 1.252 & 1.259 \\\\\n$9.00$--$9.50$ & 1.317 & 1.303 & 1.287 & 1.273 & 1.267 & 1.256 & 1.249 & 1.250 \\\\\n$9.50$--$10.00$ & 1.310 & 1.301 & 1.286 & 1.275 & 1.262 & 1.255 & 1.248 & 1.247 \\\\\n$10.00$--$10.50$ & 1.303 & 1.294 & 1.283 & 1.271 & 1.265 & 1.257 & 1.248 & 1.243 \\\\\n$10.50$--$11.00$ & 1.295 & 1.289 & 1.279 & 1.271 & 1.259 & 1.254 & 1.246 & 1.240 \\\\\n$11.00$--$11.50$ & 1.289 & 1.282 & 1.273 & 1.264 & 1.254 & 1.249 & 1.242 & 1.238 \\\\\n$11.50$--$12.00$ & 1.282 & 1.276 & 1.267 & 1.260 & 1.249 & 1.246 & 1.240 & 1.234 \\\\\n$12.00$--$13.00$ & 1.271 & 1.266 & 1.259 & 1.250 & 1.244 & 1.241 & 1.236 & 1.232 \\\\\n$13.00$--$14.00$ & 1.258 & 1.252 & 1.246 & 1.239 & 1.234 & 1.232 & 1.229 & 1.226 \\\\\n$14.00$--$15.00$ & 1.244 & 1.240 & 1.237 & 1.230 & 1.228 & 1.221 & 1.219 & 1.216 \\\\\n$15.00$--$16.00$ & 1.234 & 1.229 & 1.224 & 1.221 & 1.216 & 1.214 & 1.211 & 1.211 \\\\\n$16.00$--$17.00$ & 1.220 & 1.217 & 1.213 & 1.211 & 1.208 & 1.205 & 1.205 & 1.202 \\\\\n$17.00$--$18.00$ & 1.211 & 1.210 & 1.205 & 1.202 & 1.198 & 1.197 & 1.193 & 1.193 \\\\\n$18.00$--$20.00$ & 1.197 & 1.194 & 1.191 & 1.190 & 1.188 & 1.187 & 1.186 & 1.181 \\\\\n$20.00$--$22.00$ & 1.181 & 1.180 & 1.176 & 1.174 & 1.174 & 1.171 & 1.172 & 1.169 \\\\\n$22.00$--$24.00$ & 1.167 & 1.163 & 1.164 & 1.161 & 1.160 & 1.164 & 1.155 & 1.159 \\\\\n$24.00$--$26.00$ & 1.153 & 1.154 & 1.149 & 1.148 & 1.149 & 1.145 & 1.147 & 1.147 \\\\\n$26.00$--$30.00$ & 1.137 & 1.139 & 1.138 & 1.135 & 1.136 & 1.134 & 1.135 & 1.133 \\\\\n$30.00$--$40.00$ & 1.115 & 1.115 & 1.115 & 1.115 & 1.116 & 1.114 & 1.116 & 1.116 \\\\\n$40.00$--$60.00$ & 1.086 & 1.085 & 1.083 & 1.086 & 1.083 & 1.084 & 1.089 & 1.028 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trp0_psi2s} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``transverse positive'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 2.009 & 2.007 & 1.986 & 1.994 & 1.964 & 1.936 & 1.949 & 1.967 \\\\\n$8.50$--$9.00$ & 1.614 & 1.617 & 1.617 & 1.613 & 1.618 & 1.624 & 1.606 & 1.872 \\\\\n$9.00$--$9.50$ & 1.504 & 1.502 & 1.496 & 1.493 & 1.500 & 1.499 & 1.494 & 1.741 \\\\\n$9.50$--$10.00$ & 1.445 & 1.443 & 1.440 & 1.445 & 1.440 & 1.441 & 1.436 & 1.621 \\\\\n$10.00$--$10.50$ & 1.404 & 1.401 & 1.403 & 1.401 & 1.412 & 1.406 & 1.400 & 1.507 \\\\\n$10.50$--$11.00$ & 1.374 & 1.377 & 1.378 & 1.378 & 1.377 & 1.378 & 1.372 & 1.447 \\\\\n$11.00$--$11.50$ & 1.355 & 1.352 & 1.352 & 1.353 & 1.352 & 1.353 & 1.350 & 1.409 \\\\\n$11.50$--$12.00$ & 1.335 & 1.334 & 1.335 & 1.335 & 1.332 & 1.336 & 1.331 & 1.375 \\\\\n$12.00$--$13.00$ & 1.314 & 1.313 & 1.312 & 1.312 & 1.313 & 1.314 & 1.312 & 1.343 \\\\\n$13.00$--$14.00$ & 1.289 & 1.289 & 1.287 & 1.286 & 1.286 & 1.288 & 1.288 & 1.311 \\\\\n$14.00$--$15.00$ & 1.268 & 1.267 & 1.269 & 1.267 & 1.269 & 1.265 & 1.265 & 1.280 \\\\\n$15.00$--$16.00$ & 1.253 & 1.250 & 1.249 & 1.252 & 1.250 & 1.250 & 1.248 & 1.262 \\\\\n$16.00$--$17.00$ & 1.234 & 1.234 & 1.234 & 1.234 & 1.235 & 1.235 & 1.236 & 1.241 \\\\\n$17.00$--$18.00$ & 1.224 & 1.224 & 1.222 & 1.222 & 1.220 & 1.222 & 1.218 & 1.224 \\\\\n$18.00$--$20.00$ & 1.206 & 1.205 & 1.204 & 1.205 & 1.205 & 1.207 & 1.206 & 1.204 \\\\\n$20.00$--$22.00$ & 1.187 & 1.187 & 1.186 & 1.184 & 1.186 & 1.184 & 1.186 & 1.186 \\\\\n$22.00$--$24.00$ & 1.171 & 1.169 & 1.171 & 1.169 & 1.169 & 1.174 & 1.166 & 1.170 \\\\\n$24.00$--$26.00$ & 1.157 & 1.158 & 1.155 & 1.155 & 1.156 & 1.153 & 1.156 & 1.156 \\\\\n$26.00$--$30.00$ & 1.140 & 1.142 & 1.142 & 1.139 & 1.141 & 1.140 & 1.141 & 1.139 \\\\\n$30.00$--$40.00$ & 1.116 & 1.117 & 1.117 & 1.117 & 1.119 & 1.117 & 1.119 & 1.119 \\\\\n$40.00$--$60.00$ & 1.087 & 1.086 & 1.084 & 1.087 & 1.085 & 1.085 & 1.091 & 1.029 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trpp_psi2s} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``transverse negative'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 0.998 & 0.986 & 0.970 & 0.957 & 0.949 & 0.941 & 0.935 & 0.883 \\\\\n$8.50$--$9.00$ & 1.115 & 1.102 & 1.084 & 1.062 & 1.047 & 1.039 & 1.025 & 0.959 \\\\\n$9.00$--$9.50$ & 1.169 & 1.154 & 1.131 & 1.110 & 1.096 & 1.084 & 1.075 & 1.007 \\\\\n$9.50$--$10.00$ & 1.200 & 1.185 & 1.163 & 1.144 & 1.126 & 1.114 & 1.105 & 1.047 \\\\\n$10.00$--$10.50$ & 1.216 & 1.200 & 1.181 & 1.161 & 1.148 & 1.137 & 1.127 & 1.075 \\\\\n$10.50$--$11.00$ & 1.222 & 1.212 & 1.196 & 1.178 & 1.161 & 1.152 & 1.143 & 1.097 \\\\\n$11.00$--$11.50$ & 1.230 & 1.218 & 1.202 & 1.185 & 1.169 & 1.161 & 1.152 & 1.112 \\\\\n$11.50$--$12.00$ & 1.233 & 1.221 & 1.205 & 1.192 & 1.175 & 1.169 & 1.160 & 1.124 \\\\\n$12.00$--$13.00$ & 1.232 & 1.222 & 1.208 & 1.195 & 1.184 & 1.176 & 1.169 & 1.141 \\\\\n$13.00$--$14.00$ & 1.228 & 1.220 & 1.208 & 1.196 & 1.187 & 1.181 & 1.176 & 1.155 \\\\\n$14.00$--$15.00$ & 1.221 & 1.214 & 1.207 & 1.196 & 1.188 & 1.181 & 1.176 & 1.159 \\\\\n$15.00$--$16.00$ & 1.215 & 1.208 & 1.200 & 1.193 & 1.184 & 1.181 & 1.175 & 1.165 \\\\\n$16.00$--$17.00$ & 1.205 & 1.200 & 1.194 & 1.187 & 1.182 & 1.177 & 1.175 & 1.165 \\\\\n$17.00$--$18.00$ & 1.199 & 1.196 & 1.188 & 1.183 & 1.177 & 1.174 & 1.169 & 1.162 \\\\\n$18.00$--$20.00$ & 1.188 & 1.184 & 1.179 & 1.175 & 1.170 & 1.168 & 1.166 & 1.158 \\\\\n$20.00$--$22.00$ & 1.174 & 1.172 & 1.167 & 1.162 & 1.161 & 1.157 & 1.157 & 1.153 \\\\\n$22.00$--$24.00$ & 1.162 & 1.157 & 1.158 & 1.153 & 1.151 & 1.153 & 1.145 & 1.146 \\\\\n$24.00$--$26.00$ & 1.149 & 1.149 & 1.144 & 1.142 & 1.142 & 1.138 & 1.139 & 1.138 \\\\\n$26.00$--$30.00$ & 1.135 & 1.136 & 1.134 & 1.130 & 1.131 & 1.129 & 1.129 & 1.127 \\\\\n$30.00$--$40.00$ & 1.114 & 1.113 & 1.113 & 1.112 & 1.113 & 1.110 & 1.112 & 1.112 \\\\\n$40.00$--$60.00$ & 1.086 & 1.085 & 1.083 & 1.085 & 1.082 & 1.082 & 1.088 & 1.028 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trpm_psi2s} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``off-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane positive'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 1.017 & 1.052 & 1.081 & 1.100 & 1.118 & 1.123 & 1.129 & 1.106 \\\\\n$8.50$--$9.00$ & 1.023 & 1.064 & 1.094 & 1.118 & 1.136 & 1.146 & 1.151 & 1.132 \\\\\n$9.00$--$9.50$ & 1.021 & 1.062 & 1.093 & 1.119 & 1.140 & 1.150 & 1.153 & 1.139 \\\\\n$9.50$--$10.00$ & 1.019 & 1.060 & 1.092 & 1.119 & 1.135 & 1.144 & 1.152 & 1.146 \\\\\n$10.00$--$10.50$ & 1.020 & 1.057 & 1.088 & 1.112 & 1.132 & 1.140 & 1.146 & 1.145 \\\\\n$10.50$--$11.00$ & 1.017 & 1.055 & 1.085 & 1.108 & 1.124 & 1.134 & 1.139 & 1.141 \\\\\n$11.00$--$11.50$ & 1.017 & 1.052 & 1.079 & 1.102 & 1.118 & 1.127 & 1.131 & 1.137 \\\\\n$11.50$--$12.00$ & 1.017 & 1.050 & 1.076 & 1.096 & 1.110 & 1.120 & 1.126 & 1.130 \\\\\n$12.00$--$13.00$ & 1.014 & 1.044 & 1.069 & 1.088 & 1.102 & 1.111 & 1.116 & 1.123 \\\\\n$13.00$--$14.00$ & 1.013 & 1.041 & 1.061 & 1.078 & 1.091 & 1.100 & 1.104 & 1.111 \\\\\n$14.00$--$15.00$ & 1.012 & 1.036 & 1.056 & 1.070 & 1.082 & 1.088 & 1.092 & 1.098 \\\\\n$15.00$--$16.00$ & 1.011 & 1.032 & 1.049 & 1.064 & 1.073 & 1.079 & 1.083 & 1.090 \\\\\n$16.00$--$17.00$ & 1.010 & 1.029 & 1.045 & 1.057 & 1.065 & 1.072 & 1.076 & 1.080 \\\\\n$17.00$--$18.00$ & 1.010 & 1.027 & 1.041 & 1.051 & 1.059 & 1.064 & 1.068 & 1.071 \\\\\n$18.00$--$20.00$ & 1.008 & 1.023 & 1.035 & 1.045 & 1.052 & 1.057 & 1.059 & 1.062 \\\\\n$20.00$--$22.00$ & 1.008 & 1.019 & 1.030 & 1.037 & 1.044 & 1.047 & 1.050 & 1.052 \\\\\n$22.00$--$24.00$ & 1.006 & 1.016 & 1.025 & 1.032 & 1.037 & 1.042 & 1.042 & 1.044 \\\\\n$24.00$--$26.00$ & 1.005 & 1.014 & 1.021 & 1.027 & 1.032 & 1.034 & 1.037 & 1.038 \\\\\n$26.00$--$30.00$ & 1.005 & 1.012 & 1.018 & 1.022 & 1.027 & 1.029 & 1.030 & 1.031 \\\\\n$30.00$--$40.00$ & 1.003 & 1.008 & 1.013 & 1.016 & 1.019 & 1.020 & 1.022 & 1.022 \\\\\n$40.00$--$60.00$ & 1.002 & 1.005 & 1.007 & 1.009 & 1.010 & 1.011 & 1.013 & 1.004 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_offP_psi2s} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``off-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane negative'' spin-alignment hypothesis for 7 \\TeV.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n$8.00$--$8.50$ & 0.983 & 0.950 & 0.931 & 0.916 & 0.908 & 0.902 & 0.902 & 0.911 \\\\\n$8.50$--$9.00$ & 0.979 & 0.944 & 0.919 & 0.904 & 0.892 & 0.887 & 0.882 & 0.898 \\\\\n$9.00$--$9.50$ & 0.981 & 0.943 & 0.919 & 0.901 & 0.894 & 0.886 & 0.883 & 0.891 \\\\\n$9.50$--$10.00$ & 0.981 & 0.945 & 0.922 & 0.903 & 0.894 & 0.886 & 0.885 & 0.891 \\\\\n$10.00$--$10.50$ & 0.982 & 0.948 & 0.925 & 0.910 & 0.897 & 0.891 & 0.888 & 0.890 \\\\\n$10.50$--$11.00$ & 0.982 & 0.951 & 0.929 & 0.913 & 0.901 & 0.895 & 0.891 & 0.892 \\\\\n$11.00$--$11.50$ & 0.983 & 0.953 & 0.931 & 0.918 & 0.906 & 0.900 & 0.897 & 0.894 \\\\\n$11.50$--$12.00$ & 0.985 & 0.955 & 0.934 & 0.920 & 0.910 & 0.903 & 0.901 & 0.897 \\\\\n$12.00$--$13.00$ & 0.985 & 0.958 & 0.938 & 0.925 & 0.915 & 0.911 & 0.906 & 0.903 \\\\\n$13.00$--$14.00$ & 0.988 & 0.963 & 0.944 & 0.932 & 0.924 & 0.918 & 0.915 & 0.910 \\\\\n$14.00$--$15.00$ & 0.988 & 0.966 & 0.950 & 0.939 & 0.930 & 0.926 & 0.923 & 0.918 \\\\\n$15.00$--$16.00$ & 0.990 & 0.969 & 0.955 & 0.944 & 0.937 & 0.932 & 0.929 & 0.924 \\\\\n$16.00$--$17.00$ & 0.991 & 0.972 & 0.959 & 0.949 & 0.941 & 0.937 & 0.935 & 0.932 \\\\\n$17.00$--$18.00$ & 0.991 & 0.975 & 0.963 & 0.953 & 0.947 & 0.942 & 0.941 & 0.938 \\\\\n$18.00$--$20.00$ & 0.992 & 0.978 & 0.967 & 0.959 & 0.953 & 0.949 & 0.947 & 0.945 \\\\\n$20.00$--$22.00$ & 0.993 & 0.981 & 0.972 & 0.965 & 0.960 & 0.957 & 0.955 & 0.953 \\\\\n$22.00$--$24.00$ & 0.994 & 0.984 & 0.975 & 0.970 & 0.966 & 0.962 & 0.961 & 0.959 \\\\\n$24.00$--$26.00$ & 0.995 & 0.986 & 0.979 & 0.974 & 0.970 & 0.968 & 0.966 & 0.965 \\\\\n$26.00$--$30.00$ & 0.995 & 0.988 & 0.983 & 0.978 & 0.975 & 0.973 & 0.972 & 0.971 \\\\\n$30.00$--$40.00$ & 0.997 & 0.992 & 0.988 & 0.984 & 0.982 & 0.981 & 0.979 & 0.979 \\\\\n$40.00$--$60.00$ & 0.998 & 0.995 & 0.993 & 0.991 & 0.990 & 0.989 & 0.988 & 0.996 \\\\\n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_offN_psi2s} \n \\end{table}\n\n\n\\clearpage\n\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``longitudinal'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here. } \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 0.672 & 0.674 & 0.678 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 0.670 & 0.673 & 0.678 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 0.671 & 0.674 & 0.679 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 0.674 & 0.676 & 0.681 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 0.676 & 0.678 & 0.683 & 0.686 & 0.691 & 0.694 & 0.695 & 0.696 \\\\ \n 10.50 -- 11.00 & 0.680 & 0.681 & 0.686 & 0.689 & 0.693 & 0.696 & 0.697 & 0.698 \\\\ \n 11.00 -- 11.50 & 0.684 & 0.685 & 0.690 & 0.692 & 0.695 & 0.698 & 0.700 & 0.701 \\\\ \n 11.50 -- 12.00 & 0.688 & 0.688 & 0.693 & 0.695 & 0.698 & 0.701 & 0.702 & 0.704 \\\\ \n 12.00 -- 12.50 & 0.692 & 0.692 & 0.696 & 0.698 & 0.702 & 0.704 & 0.705 & 0.706 \\\\ \n 12.50 -- 13.00 & 0.696 & 0.696 & 0.700 & 0.702 & 0.705 & 0.707 & 0.708 & 0.710 \\\\ \n 13.00 -- 14.00 & 0.702 & 0.703 & 0.705 & 0.707 & 0.710 & 0.712 & 0.713 & 0.715 \\\\ \n 14.00 -- 15.00 & 0.710 & 0.711 & 0.713 & 0.714 & 0.717 & 0.719 & 0.720 & 0.722 \\\\ \n 15.00 -- 16.00 & 0.719 & 0.719 & 0.721 & 0.722 & 0.724 & 0.725 & 0.727 & 0.729 \\\\ \n 16.00 -- 17.00 & 0.726 & 0.727 & 0.729 & 0.729 & 0.732 & 0.733 & 0.734 & 0.735 \\\\ \n 17.00 -- 18.00 & 0.734 & 0.735 & 0.736 & 0.737 & 0.738 & 0.740 & 0.740 & 0.743 \\\\ \n 18.00 -- 20.00 & 0.744 & 0.745 & 0.746 & 0.746 & 0.748 & 0.750 & 0.750 & 0.752 \\\\ \n 20.00 -- 22.00 & 0.758 & 0.759 & 0.760 & 0.759 & 0.761 & 0.762 & 0.763 & 0.764 \\\\ \n 22.00 -- 24.00 & 0.771 & 0.771 & 0.772 & 0.771 & 0.773 & 0.774 & 0.774 & 0.776 \\\\ \n 24.00 -- 26.00 & 0.783 & 0.783 & 0.783 & 0.783 & 0.784 & 0.786 & 0.786 & 0.787 \\\\ \n 26.00 -- 30.00 & 0.797 & 0.798 & 0.798 & 0.797 & 0.798 & 0.799 & 0.800 & 0.800 \\\\ \n 30.00 -- 35.00 & 0.817 & 0.817 & 0.817 & 0.816 & 0.817 & 0.818 & 0.818 & 0.820 \\\\ \n 35.00 -- 40.00 & 0.836 & 0.836 & 0.836 & 0.835 & 0.835 & 0.836 & 0.836 & 0.840 \\\\ \n 40.00 -- 60.00 & 0.862 & 0.862 & 0.861 & 0.861 & 0.861 & 0.862 & 0.862 & 0.863 \\\\ \n 60.00 -- 110.00 & 0.904 & 0.902 & 0.903 & 0.902 & 0.903 & 0.904 & 0.905 & 0.906 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_long_jpsi8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``transverse zero'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 1.326 & 1.321 & 1.311 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 1.326 & 1.320 & 1.309 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 1.322 & 1.316 & 1.306 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 1.317 & 1.312 & 1.302 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 1.311 & 1.306 & 1.297 & 1.291 & 1.283 & 1.278 & 1.275 & 1.273 \\\\ \n 10.50 -- 11.00 & 1.304 & 1.300 & 1.292 & 1.286 & 1.279 & 1.274 & 1.272 & 1.269 \\\\ \n 11.00 -- 11.50 & 1.297 & 1.293 & 1.286 & 1.280 & 1.275 & 1.270 & 1.268 & 1.265 \\\\ \n 11.50 -- 12.00 & 1.290 & 1.287 & 1.280 & 1.275 & 1.270 & 1.266 & 1.263 & 1.261 \\\\ \n 12.00 -- 12.50 & 1.283 & 1.280 & 1.274 & 1.270 & 1.264 & 1.261 & 1.259 & 1.257 \\\\ \n 12.50 -- 13.00 & 1.276 & 1.273 & 1.268 & 1.264 & 1.260 & 1.256 & 1.254 & 1.252 \\\\ \n 13.00 -- 14.00 & 1.265 & 1.264 & 1.259 & 1.256 & 1.252 & 1.249 & 1.247 & 1.245 \\\\ \n 14.00 -- 15.00 & 1.253 & 1.251 & 1.247 & 1.245 & 1.241 & 1.238 & 1.237 & 1.235 \\\\ \n 15.00 -- 16.00 & 1.240 & 1.239 & 1.236 & 1.234 & 1.231 & 1.229 & 1.227 & 1.225 \\\\ \n 16.00 -- 17.00 & 1.228 & 1.227 & 1.225 & 1.223 & 1.220 & 1.218 & 1.218 & 1.216 \\\\ \n 17.00 -- 18.00 & 1.218 & 1.217 & 1.215 & 1.213 & 1.211 & 1.209 & 1.209 & 1.206 \\\\ \n 18.00 -- 20.00 & 1.204 & 1.203 & 1.201 & 1.201 & 1.199 & 1.197 & 1.196 & 1.195 \\\\ \n 20.00 -- 22.00 & 1.186 & 1.186 & 1.185 & 1.185 & 1.183 & 1.182 & 1.181 & 1.180 \\\\ \n 22.00 -- 24.00 & 1.172 & 1.171 & 1.171 & 1.171 & 1.169 & 1.168 & 1.168 & 1.167 \\\\ \n 24.00 -- 26.00 & 1.159 & 1.159 & 1.158 & 1.158 & 1.157 & 1.156 & 1.156 & 1.154 \\\\ \n 26.00 -- 30.00 & 1.144 & 1.144 & 1.143 & 1.144 & 1.143 & 1.142 & 1.141 & 1.141 \\\\ \n 30.00 -- 35.00 & 1.125 & 1.125 & 1.125 & 1.125 & 1.124 & 1.124 & 1.124 & 1.122 \\\\ \n 35.00 -- 40.00 & 1.108 & 1.108 & 1.108 & 1.108 & 1.108 & 1.108 & 1.107 & 1.105 \\\\ \n 40.00 -- 60.00 & 1.087 & 1.086 & 1.087 & 1.087 & 1.087 & 1.087 & 1.087 & 1.086 \\\\ \n 60.00 -- 110.00 & 1.056 & 1.057 & 1.057 & 1.057 & 1.057 & 1.056 & 1.055 & 1.055 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trp0_jpsi8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``transverse positive'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 1.926 & 1.933 & 1.930 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 1.555 & 1.558 & 1.559 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 1.463 & 1.464 & 1.465 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 1.416 & 1.418 & 1.418 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 1.386 & 1.388 & 1.387 & 1.390 & 1.390 & 1.390 & 1.391 & 1.411 \\\\ \n 10.50 -- 11.00 & 1.363 & 1.365 & 1.365 & 1.367 & 1.367 & 1.366 & 1.368 & 1.382 \\\\ \n 11.00 -- 11.50 & 1.345 & 1.347 & 1.346 & 1.348 & 1.348 & 1.348 & 1.349 & 1.358 \\\\ \n 11.50 -- 12.00 & 1.330 & 1.331 & 1.331 & 1.333 & 1.333 & 1.332 & 1.333 & 1.340 \\\\ \n 12.00 -- 12.50 & 1.316 & 1.318 & 1.317 & 1.319 & 1.318 & 1.319 & 1.319 & 1.325 \\\\ \n 12.50 -- 13.00 & 1.304 & 1.305 & 1.305 & 1.307 & 1.307 & 1.307 & 1.306 & 1.311 \\\\ \n 13.00 -- 14.00 & 1.288 & 1.290 & 1.290 & 1.291 & 1.291 & 1.291 & 1.291 & 1.293 \\\\ \n 14.00 -- 15.00 & 1.270 & 1.271 & 1.271 & 1.272 & 1.272 & 1.271 & 1.272 & 1.272 \\\\ \n 15.00 -- 16.00 & 1.253 & 1.254 & 1.254 & 1.255 & 1.255 & 1.255 & 1.254 & 1.255 \\\\ \n 16.00 -- 17.00 & 1.239 & 1.240 & 1.240 & 1.241 & 1.240 & 1.240 & 1.241 & 1.240 \\\\ \n 17.00 -- 18.00 & 1.227 & 1.227 & 1.227 & 1.228 & 1.228 & 1.227 & 1.228 & 1.226 \\\\ \n 18.00 -- 20.00 & 1.211 & 1.211 & 1.211 & 1.212 & 1.212 & 1.211 & 1.211 & 1.210 \\\\ \n 20.00 -- 22.00 & 1.191 & 1.192 & 1.192 & 1.193 & 1.193 & 1.192 & 1.192 & 1.192 \\\\ \n 22.00 -- 24.00 & 1.175 & 1.176 & 1.176 & 1.177 & 1.176 & 1.176 & 1.176 & 1.175 \\\\ \n 24.00 -- 26.00 & 1.162 & 1.162 & 1.162 & 1.163 & 1.162 & 1.162 & 1.162 & 1.161 \\\\ \n 26.00 -- 30.00 & 1.146 & 1.146 & 1.146 & 1.147 & 1.146 & 1.146 & 1.146 & 1.146 \\\\ \n 30.00 -- 35.00 & 1.126 & 1.126 & 1.126 & 1.127 & 1.127 & 1.126 & 1.126 & 1.125 \\\\ \n 35.00 -- 40.00 & 1.109 & 1.109 & 1.109 & 1.110 & 1.110 & 1.109 & 1.109 & 1.107 \\\\ \n 40.00 -- 60.00 & 1.087 & 1.087 & 1.088 & 1.088 & 1.088 & 1.087 & 1.087 & 1.087 \\\\ \n 60.00 -- 110.00 & 1.056 & 1.057 & 1.057 & 1.057 & 1.057 & 1.056 & 1.056 & 1.055 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trpp_jpsi8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``transverse negative'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 1.026 & 1.017 & 1.005 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 1.157 & 1.145 & 1.129 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 1.207 & 1.196 & 1.178 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 1.231 & 1.220 & 1.203 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 1.244 & 1.234 & 1.218 & 1.204 & 1.192 & 1.182 & 1.177 & 1.161 \\\\ \n 10.50 -- 11.00 & 1.250 & 1.241 & 1.227 & 1.214 & 1.202 & 1.193 & 1.188 & 1.175 \\\\ \n 11.00 -- 11.50 & 1.252 & 1.244 & 1.231 & 1.220 & 1.209 & 1.200 & 1.195 & 1.184 \\\\ \n 11.50 -- 12.00 & 1.253 & 1.246 & 1.234 & 1.223 & 1.213 & 1.206 & 1.201 & 1.191 \\\\ \n 12.00 -- 12.50 & 1.251 & 1.245 & 1.234 & 1.224 & 1.215 & 1.208 & 1.204 & 1.196 \\\\ \n 12.50 -- 13.00 & 1.248 & 1.243 & 1.233 & 1.224 & 1.216 & 1.210 & 1.206 & 1.199 \\\\ \n 13.00 -- 14.00 & 1.243 & 1.239 & 1.230 & 1.222 & 1.215 & 1.210 & 1.206 & 1.200 \\\\ \n 14.00 -- 15.00 & 1.236 & 1.231 & 1.224 & 1.218 & 1.212 & 1.207 & 1.204 & 1.200 \\\\ \n 15.00 -- 16.00 & 1.226 & 1.223 & 1.217 & 1.212 & 1.207 & 1.203 & 1.200 & 1.197 \\\\ \n 16.00 -- 17.00 & 1.218 & 1.215 & 1.210 & 1.206 & 1.201 & 1.197 & 1.195 & 1.193 \\\\ \n 17.00 -- 18.00 & 1.209 & 1.206 & 1.202 & 1.199 & 1.195 & 1.192 & 1.190 & 1.187 \\\\ \n 18.00 -- 20.00 & 1.197 & 1.195 & 1.192 & 1.189 & 1.186 & 1.183 & 1.182 & 1.180 \\\\ \n 20.00 -- 22.00 & 1.182 & 1.181 & 1.178 & 1.177 & 1.174 & 1.172 & 1.170 & 1.170 \\\\ \n 22.00 -- 24.00 & 1.168 & 1.167 & 1.166 & 1.165 & 1.162 & 1.161 & 1.160 & 1.159 \\\\ \n 24.00 -- 26.00 & 1.156 & 1.156 & 1.154 & 1.153 & 1.152 & 1.150 & 1.150 & 1.148 \\\\ \n 26.00 -- 30.00 & 1.142 & 1.141 & 1.140 & 1.140 & 1.139 & 1.137 & 1.137 & 1.136 \\\\ \n 30.00 -- 35.00 & 1.124 & 1.123 & 1.123 & 1.123 & 1.122 & 1.121 & 1.121 & 1.119 \\\\ \n 35.00 -- 40.00 & 1.107 & 1.107 & 1.107 & 1.107 & 1.107 & 1.106 & 1.106 & 1.103 \\\\ \n 40.00 -- 60.00 & 1.087 & 1.086 & 1.087 & 1.086 & 1.087 & 1.086 & 1.086 & 1.085 \\\\ \n 60.00 -- 110.00 & 1.056 & 1.057 & 1.057 & 1.057 & 1.056 & 1.056 & 1.055 & 1.055 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trpm_jpsi8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``off-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane positive'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 1.016 & 1.048 & 1.074 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 1.019 & 1.056 & 1.087 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 1.019 & 1.055 & 1.086 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 1.018 & 1.053 & 1.083 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 1.017 & 1.051 & 1.079 & 1.101 & 1.117 & 1.127 & 1.134 & 1.138 \\\\ \n 10.50 -- 11.00 & 1.016 & 1.048 & 1.075 & 1.096 & 1.110 & 1.120 & 1.126 & 1.131 \\\\ \n 11.00 -- 11.50 & 1.015 & 1.045 & 1.071 & 1.090 & 1.104 & 1.113 & 1.119 & 1.124 \\\\ \n 11.50 -- 12.00 & 1.014 & 1.043 & 1.067 & 1.085 & 1.098 & 1.107 & 1.113 & 1.117 \\\\ \n 12.00 -- 12.50 & 1.014 & 1.040 & 1.063 & 1.080 & 1.093 & 1.101 & 1.106 & 1.111 \\\\ \n 12.50 -- 13.00 & 1.013 & 1.038 & 1.059 & 1.076 & 1.087 & 1.095 & 1.100 & 1.104 \\\\ \n 13.00 -- 14.00 & 1.012 & 1.035 & 1.055 & 1.070 & 1.080 & 1.088 & 1.092 & 1.096 \\\\ \n 14.00 -- 15.00 & 1.011 & 1.031 & 1.049 & 1.062 & 1.072 & 1.078 & 1.082 & 1.085 \\\\ \n 15.00 -- 16.00 & 1.010 & 1.028 & 1.044 & 1.056 & 1.065 & 1.070 & 1.074 & 1.076 \\\\ \n 16.00 -- 17.00 & 1.009 & 1.025 & 1.040 & 1.050 & 1.058 & 1.063 & 1.067 & 1.069 \\\\ \n 17.00 -- 18.00 & 1.008 & 1.023 & 1.036 & 1.046 & 1.053 & 1.057 & 1.060 & 1.062 \\\\ \n 18.00 -- 20.00 & 1.007 & 1.020 & 1.031 & 1.040 & 1.046 & 1.050 & 1.053 & 1.054 \\\\ \n 20.00 -- 22.00 & 1.006 & 1.017 & 1.026 & 1.033 & 1.039 & 1.042 & 1.044 & 1.045 \\\\ \n 22.00 -- 24.00 & 1.005 & 1.014 & 1.022 & 1.028 & 1.033 & 1.036 & 1.038 & 1.039 \\\\ \n 24.00 -- 26.00 & 1.004 & 1.012 & 1.019 & 1.024 & 1.028 & 1.030 & 1.032 & 1.033 \\\\ \n 26.00 -- 30.00 & 1.004 & 1.010 & 1.016 & 1.020 & 1.023 & 1.025 & 1.026 & 1.027 \\\\ \n 30.00 -- 35.00 & 1.003 & 1.008 & 1.012 & 1.015 & 1.018 & 1.019 & 1.020 & 1.021 \\\\ \n 35.00 -- 40.00 & 1.002 & 1.006 & 1.009 & 1.012 & 1.013 & 1.015 & 1.015 & 1.015 \\\\ \n 40.00 -- 60.00 & 1.001 & 1.004 & 1.006 & 1.008 & 1.009 & 1.010 & 1.010 & 1.010 \\\\ \n 60.00 -- 110.00 & 1.001 & 1.002 & 1.003 & 1.003 & 1.004 & 1.004 & 1.004 & 1.005 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_offP_jpsi8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\jpsi$ under the ``off-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane negative'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 0.985 & 0.957 & 0.936 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 0.982 & 0.950 & 0.926 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 0.982 & 0.950 & 0.926 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 0.983 & 0.952 & 0.929 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 0.984 & 0.954 & 0.932 & 0.916 & 0.905 & 0.898 & 0.894 & 0.891 \\\\ \n 10.50 -- 11.00 & 0.985 & 0.956 & 0.935 & 0.919 & 0.909 & 0.903 & 0.899 & 0.895 \\\\ \n 11.00 -- 11.50 & 0.985 & 0.959 & 0.938 & 0.923 & 0.913 & 0.907 & 0.903 & 0.900 \\\\ \n 11.50 -- 12.00 & 0.986 & 0.961 & 0.941 & 0.927 & 0.918 & 0.911 & 0.908 & 0.905 \\\\ \n 12.00 -- 12.50 & 0.987 & 0.963 & 0.944 & 0.931 & 0.922 & 0.916 & 0.912 & 0.909 \\\\ \n 12.50 -- 13.00 & 0.988 & 0.965 & 0.947 & 0.934 & 0.925 & 0.920 & 0.916 & 0.913 \\\\ \n 13.00 -- 14.00 & 0.988 & 0.967 & 0.951 & 0.939 & 0.930 & 0.925 & 0.922 & 0.919 \\\\ \n 14.00 -- 15.00 & 0.990 & 0.971 & 0.955 & 0.944 & 0.937 & 0.932 & 0.929 & 0.927 \\\\ \n 15.00 -- 16.00 & 0.991 & 0.974 & 0.960 & 0.950 & 0.943 & 0.938 & 0.936 & 0.934 \\\\ \n 16.00 -- 17.00 & 0.991 & 0.976 & 0.963 & 0.954 & 0.948 & 0.944 & 0.941 & 0.939 \\\\ \n 17.00 -- 18.00 & 0.992 & 0.978 & 0.967 & 0.958 & 0.952 & 0.949 & 0.946 & 0.945 \\\\ \n 18.00 -- 20.00 & 0.993 & 0.981 & 0.971 & 0.963 & 0.958 & 0.954 & 0.952 & 0.951 \\\\ \n 20.00 -- 22.00 & 0.994 & 0.984 & 0.975 & 0.969 & 0.964 & 0.961 & 0.959 & 0.958 \\\\ \n 22.00 -- 24.00 & 0.995 & 0.986 & 0.979 & 0.973 & 0.969 & 0.967 & 0.965 & 0.964 \\\\ \n 24.00 -- 26.00 & 0.996 & 0.988 & 0.982 & 0.977 & 0.973 & 0.971 & 0.970 & 0.969 \\\\ \n 26.00 -- 30.00 & 0.996 & 0.990 & 0.985 & 0.981 & 0.978 & 0.976 & 0.975 & 0.974 \\\\ \n 30.00 -- 35.00 & 0.997 & 0.992 & 0.988 & 0.985 & 0.983 & 0.982 & 0.981 & 0.980 \\\\ \n 35.00 -- 40.00 & 0.998 & 0.994 & 0.991 & 0.989 & 0.987 & 0.986 & 0.985 & 0.985 \\\\ \n 40.00 -- 60.00 & 0.999 & 0.996 & 0.994 & 0.992 & 0.991 & 0.991 & 0.990 & 0.990 \\\\ \n 60.00 -- 110.00 & 0.999 & 0.998 & 0.997 & 0.997 & 0.996 & 0.996 & 0.996 & 0.996 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_offN_jpsi8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``longitudinal'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 0.672 & 0.677 & 0.686 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 0.674 & 0.680 & 0.689 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 0.677 & 0.682 & 0.691 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 0.680 & 0.684 & 0.692 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 0.683 & 0.688 & 0.695 & 0.702 & 0.709 & 0.713 & 0.717 & 0.721 \\\\ \n 10.50 -- 11.00 & 0.687 & 0.692 & 0.698 & 0.705 & 0.710 & 0.715 & 0.718 & 0.722 \\\\ \n 11.00 -- 11.50 & 0.692 & 0.695 & 0.701 & 0.708 & 0.714 & 0.716 & 0.718 & 0.725 \\\\ \n 11.50 -- 12.00 & 0.695 & 0.698 & 0.704 & 0.710 & 0.715 & 0.718 & 0.721 & 0.725 \\\\ \n 12.00 -- 12.50 & 0.700 & 0.703 & 0.708 & 0.713 & 0.718 & 0.721 & 0.723 & 0.728 \\\\ \n 12.50 -- 13.00 & 0.704 & 0.706 & 0.711 & 0.716 & 0.721 & 0.722 & 0.726 & 0.730 \\\\ \n 13.00 -- 14.00 & 0.710 & 0.713 & 0.717 & 0.722 & 0.725 & 0.727 & 0.730 & 0.733 \\\\ \n 14.00 -- 15.00 & 0.719 & 0.721 & 0.724 & 0.728 & 0.731 & 0.733 & 0.736 & 0.738 \\\\ \n 15.00 -- 16.00 & 0.727 & 0.728 & 0.732 & 0.735 & 0.737 & 0.740 & 0.741 & 0.743 \\\\ \n 16.00 -- 17.00 & 0.735 & 0.737 & 0.739 & 0.742 & 0.743 & 0.746 & 0.748 & 0.750 \\\\ \n 17.00 -- 18.00 & 0.742 & 0.744 & 0.746 & 0.750 & 0.750 & 0.753 & 0.755 & 0.755 \\\\ \n 18.00 -- 20.00 & 0.753 & 0.754 & 0.756 & 0.759 & 0.760 & 0.761 & 0.762 & 0.765 \\\\ \n 20.00 -- 22.00 & 0.767 & 0.768 & 0.769 & 0.771 & 0.773 & 0.773 & 0.775 & 0.775 \\\\ \n 22.00 -- 24.00 & 0.779 & 0.779 & 0.782 & 0.783 & 0.784 & 0.785 & 0.785 & 0.788 \\\\ \n 24.00 -- 26.00 & 0.791 & 0.791 & 0.793 & 0.794 & 0.793 & 0.795 & 0.795 & 0.795 \\\\ \n 26.00 -- 30.00 & 0.805 & 0.804 & 0.806 & 0.807 & 0.808 & 0.809 & 0.809 & 0.811 \\\\ \n 30.00 -- 35.00 & 0.823 & 0.823 & 0.824 & 0.824 & 0.826 & 0.826 & 0.828 & 0.828 \\\\ \n 35.00 -- 40.00 & 0.841 & 0.841 & 0.840 & 0.842 & 0.843 & 0.842 & 0.843 & 0.843 \\\\ \n 40.00 -- 60.00 & 0.866 & 0.867 & 0.866 & 0.868 & 0.868 & 0.866 & 0.868 & 0.870 \\\\ \n 60.00 -- 110.00 & 0.905 & 0.906 & 0.906 & 0.909 & 0.907 & 0.903 & 0.906 & 0.905 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_long_psi2s8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``transverse zero'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 1.325 & 1.316 & 1.301 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 1.321 & 1.311 & 1.295 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 1.316 & 1.307 & 1.291 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 1.310 & 1.301 & 1.288 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 1.303 & 1.295 & 1.283 & 1.272 & 1.261 & 1.254 & 1.249 & 1.244 \\\\ \n 10.50 -- 11.00 & 1.296 & 1.289 & 1.278 & 1.267 & 1.259 & 1.252 & 1.247 & 1.241 \\\\ \n 11.00 -- 11.50 & 1.289 & 1.283 & 1.273 & 1.262 & 1.254 & 1.250 & 1.246 & 1.238 \\\\ \n 11.50 -- 12.00 & 1.282 & 1.276 & 1.267 & 1.258 & 1.251 & 1.246 & 1.242 & 1.236 \\\\ \n 12.00 -- 12.50 & 1.274 & 1.270 & 1.261 & 1.253 & 1.247 & 1.242 & 1.239 & 1.233 \\\\ \n 12.50 -- 13.00 & 1.267 & 1.263 & 1.256 & 1.248 & 1.242 & 1.239 & 1.235 & 1.230 \\\\ \n 13.00 -- 14.00 & 1.257 & 1.254 & 1.247 & 1.241 & 1.236 & 1.232 & 1.229 & 1.225 \\\\ \n 14.00 -- 15.00 & 1.244 & 1.241 & 1.236 & 1.230 & 1.227 & 1.223 & 1.220 & 1.217 \\\\ \n 15.00 -- 16.00 & 1.232 & 1.230 & 1.225 & 1.221 & 1.217 & 1.215 & 1.213 & 1.211 \\\\ \n 16.00 -- 17.00 & 1.221 & 1.218 & 1.215 & 1.211 & 1.209 & 1.206 & 1.204 & 1.202 \\\\ \n 17.00 -- 18.00 & 1.210 & 1.208 & 1.206 & 1.202 & 1.200 & 1.197 & 1.195 & 1.195 \\\\ \n 18.00 -- 20.00 & 1.197 & 1.195 & 1.193 & 1.190 & 1.188 & 1.187 & 1.186 & 1.184 \\\\ \n 20.00 -- 22.00 & 1.180 & 1.179 & 1.177 & 1.175 & 1.173 & 1.172 & 1.171 & 1.171 \\\\ \n 22.00 -- 24.00 & 1.165 & 1.165 & 1.163 & 1.162 & 1.161 & 1.159 & 1.159 & 1.157 \\\\ \n 24.00 -- 26.00 & 1.153 & 1.153 & 1.151 & 1.150 & 1.150 & 1.149 & 1.149 & 1.149 \\\\ \n 26.00 -- 30.00 & 1.138 & 1.139 & 1.138 & 1.136 & 1.136 & 1.135 & 1.135 & 1.133 \\\\ \n 30.00 -- 35.00 & 1.121 & 1.121 & 1.120 & 1.119 & 1.119 & 1.118 & 1.117 & 1.117 \\\\ \n 35.00 -- 40.00 & 1.105 & 1.104 & 1.105 & 1.104 & 1.103 & 1.104 & 1.103 & 1.103 \\\\ \n 40.00 -- 60.00 & 1.084 & 1.083 & 1.084 & 1.083 & 1.083 & 1.084 & 1.083 & 1.081 \\\\ \n 60.00 -- 110.00 & 1.056 & 1.055 & 1.055 & 1.053 & 1.054 & 1.057 & 1.055 & 1.056 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trp0_psi2s8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``transverse positive'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 2.029 & 2.023 & 2.022 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 1.620 & 1.620 & 1.618 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 1.504 & 1.504 & 1.502 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 1.444 & 1.444 & 1.443 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 1.405 & 1.405 & 1.404 & 1.404 & 1.402 & 1.401 & 1.400 & 1.500 \\\\ \n 10.50 -- 11.00 & 1.377 & 1.377 & 1.376 & 1.375 & 1.375 & 1.373 & 1.373 & 1.443 \\\\ \n 11.00 -- 11.50 & 1.354 & 1.354 & 1.354 & 1.352 & 1.351 & 1.353 & 1.353 & 1.403 \\\\ \n 11.50 -- 12.00 & 1.336 & 1.336 & 1.335 & 1.334 & 1.335 & 1.334 & 1.333 & 1.375 \\\\ \n 12.00 -- 12.50 & 1.320 & 1.320 & 1.320 & 1.319 & 1.319 & 1.319 & 1.318 & 1.351 \\\\ \n 12.50 -- 13.00 & 1.306 & 1.307 & 1.306 & 1.305 & 1.304 & 1.306 & 1.304 & 1.331 \\\\ \n 13.00 -- 14.00 & 1.289 & 1.289 & 1.289 & 1.288 & 1.288 & 1.288 & 1.287 & 1.308 \\\\ \n 14.00 -- 15.00 & 1.268 & 1.269 & 1.268 & 1.267 & 1.268 & 1.267 & 1.266 & 1.281 \\\\ \n 15.00 -- 16.00 & 1.251 & 1.251 & 1.250 & 1.251 & 1.251 & 1.250 & 1.250 & 1.261 \\\\ \n 16.00 -- 17.00 & 1.236 & 1.236 & 1.236 & 1.236 & 1.236 & 1.235 & 1.235 & 1.242 \\\\ \n 17.00 -- 18.00 & 1.223 & 1.222 & 1.223 & 1.222 & 1.223 & 1.221 & 1.221 & 1.227 \\\\ \n 18.00 -- 20.00 & 1.206 & 1.206 & 1.206 & 1.205 & 1.206 & 1.206 & 1.206 & 1.208 \\\\ \n 20.00 -- 22.00 & 1.186 & 1.186 & 1.186 & 1.186 & 1.186 & 1.186 & 1.185 & 1.187 \\\\ \n 22.00 -- 24.00 & 1.170 & 1.171 & 1.170 & 1.170 & 1.170 & 1.170 & 1.170 & 1.169 \\\\ \n 24.00 -- 26.00 & 1.157 & 1.157 & 1.156 & 1.156 & 1.157 & 1.157 & 1.157 & 1.158 \\\\ \n 26.00 -- 30.00 & 1.141 & 1.142 & 1.141 & 1.141 & 1.141 & 1.140 & 1.141 & 1.140 \\\\ \n 30.00 -- 35.00 & 1.122 & 1.122 & 1.122 & 1.122 & 1.122 & 1.122 & 1.121 & 1.121 \\\\ \n 35.00 -- 40.00 & 1.106 & 1.105 & 1.106 & 1.106 & 1.105 & 1.106 & 1.105 & 1.105 \\\\ \n 40.00 -- 60.00 & 1.085 & 1.084 & 1.085 & 1.084 & 1.084 & 1.085 & 1.084 & 1.083 \\\\ \n 60.00 -- 110.00 & 1.056 & 1.055 & 1.055 & 1.054 & 1.054 & 1.057 & 1.055 & 1.056 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trpp_psi2s8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``transverse negative'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 0.995 & 0.986 & 0.970 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 1.116 & 1.102 & 1.081 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 1.170 & 1.156 & 1.133 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 1.199 & 1.185 & 1.163 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 1.215 & 1.202 & 1.182 & 1.163 & 1.146 & 1.135 & 1.127 & 1.075 \\\\ \n 10.50 -- 11.00 & 1.225 & 1.212 & 1.194 & 1.175 & 1.161 & 1.150 & 1.142 & 1.098 \\\\ \n 11.00 -- 11.50 & 1.230 & 1.218 & 1.201 & 1.184 & 1.170 & 1.161 & 1.155 & 1.114 \\\\ \n 11.50 -- 12.00 & 1.232 & 1.222 & 1.206 & 1.190 & 1.178 & 1.169 & 1.162 & 1.127 \\\\ \n 12.00 -- 12.50 & 1.232 & 1.223 & 1.208 & 1.194 & 1.182 & 1.174 & 1.168 & 1.137 \\\\ \n 12.50 -- 13.00 & 1.231 & 1.223 & 1.210 & 1.196 & 1.185 & 1.178 & 1.172 & 1.146 \\\\ \n 13.00 -- 14.00 & 1.228 & 1.220 & 1.209 & 1.197 & 1.188 & 1.181 & 1.176 & 1.154 \\\\ \n 14.00 -- 15.00 & 1.221 & 1.215 & 1.206 & 1.196 & 1.188 & 1.182 & 1.177 & 1.161 \\\\ \n 15.00 -- 16.00 & 1.214 & 1.209 & 1.200 & 1.193 & 1.186 & 1.181 & 1.177 & 1.165 \\\\ \n 16.00 -- 17.00 & 1.206 & 1.202 & 1.195 & 1.188 & 1.183 & 1.178 & 1.175 & 1.166 \\\\ \n 17.00 -- 18.00 & 1.198 & 1.195 & 1.189 & 1.183 & 1.179 & 1.174 & 1.171 & 1.165 \\\\ \n 18.00 -- 20.00 & 1.188 & 1.184 & 1.180 & 1.175 & 1.171 & 1.168 & 1.166 & 1.161 \\\\ \n 20.00 -- 22.00 & 1.173 & 1.171 & 1.168 & 1.164 & 1.161 & 1.159 & 1.157 & 1.154 \\\\ \n 22.00 -- 24.00 & 1.161 & 1.160 & 1.156 & 1.154 & 1.151 & 1.149 & 1.149 & 1.145 \\\\ \n 24.00 -- 26.00 & 1.150 & 1.149 & 1.146 & 1.144 & 1.143 & 1.141 & 1.141 & 1.140 \\\\ \n 26.00 -- 30.00 & 1.136 & 1.136 & 1.134 & 1.132 & 1.131 & 1.129 & 1.129 & 1.127 \\\\ \n 30.00 -- 35.00 & 1.119 & 1.119 & 1.117 & 1.117 & 1.115 & 1.115 & 1.113 & 1.113 \\\\ \n 35.00 -- 40.00 & 1.104 & 1.103 & 1.103 & 1.102 & 1.101 & 1.101 & 1.100 & 1.100 \\\\ \n 40.00 -- 60.00 & 1.084 & 1.083 & 1.083 & 1.082 & 1.082 & 1.083 & 1.082 & 1.080 \\\\ \n 60.00 -- 110.00 & 1.056 & 1.054 & 1.055 & 1.053 & 1.054 & 1.057 & 1.055 & 1.055 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_trpm_psi2s8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``off-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane positive'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 1.018 & 1.053 & 1.081 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 1.021 & 1.062 & 1.095 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 1.021 & 1.062 & 1.096 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 1.020 & 1.060 & 1.094 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 1.020 & 1.057 & 1.089 & 1.114 & 1.130 & 1.140 & 1.146 & 1.145 \\\\ \n 10.50 -- 11.00 & 1.018 & 1.055 & 1.085 & 1.108 & 1.124 & 1.133 & 1.139 & 1.142 \\\\ \n 11.00 -- 11.50 & 1.017 & 1.052 & 1.080 & 1.102 & 1.117 & 1.127 & 1.133 & 1.137 \\\\ \n 11.50 -- 12.00 & 1.017 & 1.049 & 1.076 & 1.096 & 1.111 & 1.120 & 1.126 & 1.132 \\\\ \n 12.00 -- 12.50 & 1.016 & 1.046 & 1.072 & 1.091 & 1.105 & 1.113 & 1.119 & 1.125 \\\\ \n 12.50 -- 13.00 & 1.015 & 1.043 & 1.068 & 1.086 & 1.099 & 1.108 & 1.112 & 1.119 \\\\ \n 13.00 -- 14.00 & 1.013 & 1.040 & 1.062 & 1.079 & 1.091 & 1.099 & 1.104 & 1.111 \\\\ \n 14.00 -- 15.00 & 1.012 & 1.036 & 1.056 & 1.071 & 1.082 & 1.089 & 1.093 & 1.099 \\\\ \n 15.00 -- 16.00 & 1.011 & 1.032 & 1.050 & 1.064 & 1.073 & 1.080 & 1.084 & 1.090 \\\\ \n 16.00 -- 17.00 & 1.010 & 1.029 & 1.045 & 1.057 & 1.067 & 1.072 & 1.076 & 1.081 \\\\ \n 17.00 -- 18.00 & 1.009 & 1.026 & 1.041 & 1.052 & 1.060 & 1.065 & 1.068 & 1.073 \\\\ \n 18.00 -- 20.00 & 1.008 & 1.023 & 1.036 & 1.045 & 1.053 & 1.057 & 1.060 & 1.063 \\\\ \n 20.00 -- 22.00 & 1.007 & 1.019 & 1.030 & 1.038 & 1.044 & 1.048 & 1.050 & 1.053 \\\\ \n 22.00 -- 24.00 & 1.006 & 1.016 & 1.025 & 1.032 & 1.037 & 1.040 & 1.043 & 1.044 \\\\ \n 24.00 -- 26.00 & 1.005 & 1.014 & 1.022 & 1.028 & 1.032 & 1.035 & 1.037 & 1.038 \\\\ \n 26.00 -- 30.00 & 1.004 & 1.012 & 1.018 & 1.023 & 1.026 & 1.029 & 1.030 & 1.031 \\\\ \n 30.00 -- 35.00 & 1.003 & 1.009 & 1.014 & 1.017 & 1.020 & 1.022 & 1.023 & 1.023 \\\\ \n 35.00 -- 40.00 & 1.002 & 1.007 & 1.010 & 1.013 & 1.015 & 1.017 & 1.017 & 1.018 \\\\ \n 40.00 -- 60.00 & 1.002 & 1.004 & 1.007 & 1.009 & 1.010 & 1.011 & 1.012 & 1.012 \\\\ \n 60.00 -- 110.00 & 1.001 & 1.002 & 1.003 & 1.004 & 1.004 & 1.005 & 1.005 & 1.005 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_offP_psi2s8} \n \\end{table}\n\n\n\\begin{table}[htp]\n \\caption{Mean weight correction factor for $\\ensuremath{\\psi(2\\mathrm{S})}$ under the ``off-($\\lambda_{\\theta}$--$\\lambda_{\\phi}$)-plane negative'' spin-alignment hypothesis for 8 \\TeV.\n Those intervals not measured in the analysis at low $\\mbox{$p_{\\text{T}}$}$, high rapidity are also excluded here.} \n \\begin{tiny} \n \\begin{center} \n \\begin{tabular}{|l|c|c|c|c|c|c|c|c|} \n \\hline \n & \\multicolumn{8}{c |}{Absolute Rapidity Range} \\\\ \\hline\n$\\mbox{$p_{\\text{T}}$}$~[\\GeV] & $0.00$--$0.25$ & $0.25$--$0.50$ & $0.50$--$0.75$ & $0.75$--$1.00$ & $1.00$--$1.25$ & $1.25$--$1.50$ & $1.50$--$1.75$ & $1.75$--$2.00$ \\\\ \\hline\n 8.00 -- 8.50 & 0.983 & 0.952 & 0.931 & -- & -- & -- & -- & -- \\\\ \n 8.50 -- 9.00 & 0.980 & 0.945 & 0.920 & -- & -- & -- & -- & -- \\\\ \n 9.00 -- 9.50 & 0.980 & 0.945 & 0.919 & -- & -- & -- & -- & -- \\\\ \n 9.50 -- 10.00 & 0.981 & 0.946 & 0.921 & -- & -- & -- & -- & -- \\\\ \n 10.00 -- 10.50 & 0.981 & 0.949 & 0.924 & 0.908 & 0.897 & 0.891 & 0.887 & 0.888 \\\\ \n 10.50 -- 11.00 & 0.982 & 0.951 & 0.928 & 0.912 & 0.901 & 0.895 & 0.891 & 0.890 \\\\ \n 11.00 -- 11.50 & 0.983 & 0.953 & 0.931 & 0.916 & 0.906 & 0.899 & 0.895 & 0.893 \\\\ \n 11.50 -- 12.00 & 0.984 & 0.956 & 0.934 & 0.919 & 0.910 & 0.903 & 0.900 & 0.896 \\\\ \n 12.00 -- 12.50 & 0.985 & 0.958 & 0.937 & 0.923 & 0.914 & 0.908 & 0.904 & 0.900 \\\\ \n 12.50 -- 13.00 & 0.986 & 0.960 & 0.940 & 0.927 & 0.918 & 0.911 & 0.908 & 0.904 \\\\ \n 13.00 -- 14.00 & 0.987 & 0.963 & 0.945 & 0.932 & 0.923 & 0.917 & 0.914 & 0.910 \\\\ \n 14.00 -- 15.00 & 0.988 & 0.967 & 0.950 & 0.938 & 0.930 & 0.925 & 0.922 & 0.917 \\\\ \n 15.00 -- 16.00 & 0.989 & 0.970 & 0.955 & 0.944 & 0.936 & 0.931 & 0.928 & 0.924 \\\\ \n 16.00 -- 17.00 & 0.990 & 0.973 & 0.959 & 0.949 & 0.941 & 0.937 & 0.934 & 0.931 \\\\ \n 17.00 -- 18.00 & 0.991 & 0.975 & 0.962 & 0.953 & 0.946 & 0.943 & 0.940 & 0.936 \\\\ \n 18.00 -- 20.00 & 0.992 & 0.978 & 0.967 & 0.958 & 0.953 & 0.949 & 0.946 & 0.944 \\\\ \n 20.00 -- 22.00 & 0.993 & 0.981 & 0.972 & 0.965 & 0.960 & 0.956 & 0.955 & 0.952 \\\\ \n 22.00 -- 24.00 & 0.994 & 0.984 & 0.976 & 0.970 & 0.965 & 0.963 & 0.961 & 0.960 \\\\ \n 24.00 -- 26.00 & 0.995 & 0.986 & 0.979 & 0.974 & 0.970 & 0.967 & 0.966 & 0.965 \\\\ \n 26.00 -- 30.00 & 0.996 & 0.989 & 0.983 & 0.978 & 0.975 & 0.973 & 0.972 & 0.971 \\\\ \n 30.00 -- 35.00 & 0.997 & 0.991 & 0.987 & 0.983 & 0.981 & 0.979 & 0.978 & 0.978 \\\\ \n 35.00 -- 40.00 & 0.998 & 0.993 & 0.990 & 0.987 & 0.985 & 0.984 & 0.983 & 0.983 \\\\ \n 40.00 -- 60.00 & 0.998 & 0.996 & 0.993 & 0.992 & 0.990 & 0.989 & 0.989 & 0.989 \\\\ \n 60.00 -- 110.00 & 0.999 & 0.998 & 0.997 & 0.996 & 0.996 & 0.995 & 0.995 & 0.995 \\\\ \n\\hline\n\\end{tabular} \n \\end{center} \n \\end{tiny} \n \\label{tab:sa_offN_psi2s8} \n \\end{table}\n\n\n\\clearpage\n\n\\section{Systematic uncertainties}\n\\label{sec:syst}\n\n\\begin{figure} [!h]\n \\begin{center}\n \\includegraphics[width=0.49\\textwidth]{hs_all_rap_0p75_1p00_jpsiP.eps}\n \\includegraphics[width=0.49\\textwidth]{hs_all_rap_0p75_1p00_jpsiNP.eps}\\\\\n \\includegraphics[width=0.49\\textwidth]{hs_all_rap_0p75_1p00_psi2sP.eps}\n \\includegraphics[width=0.49\\textwidth]{hs_all_rap_0p75_1p00_psi2sNP.eps} \n \\caption{Statistical and systematic contributions to the fractional uncertainty on the prompt (left column) and non-prompt (right column)\n $\\jpsi$ (top row) and $\\ensuremath{\\psi(2\\mathrm{S})}$ (bottom row) cross-sections for 7 \\TeV, \n shown for the region $0.75<|y|<1.00$.}\n \\label{fig:fracsyst_xs}\n \\end{center}\n\\end{figure}\n\n\\begin{figure} [!h]\n \\begin{center}\n \\includegraphics[width=0.49\\textwidth]{hs_all_rap_0p75_1p00_fNPjpsi.eps}\n \\includegraphics[width=0.49\\textwidth]{hs_all_rap_0p75_1p00_fNPpsi2s.eps}\\\\\n \\includegraphics[width=0.49\\textwidth]{hs_all_rap_0p75_1p00_ratioP.eps}\n \\includegraphics[width=0.49\\textwidth]{hs_all_rap_0p75_1p00_ratioNP.eps} \n \\caption{Breakdown of the contributions to the fractional uncertainty on the non-prompt fractions for $\\jpsi$ (top left) and $\\ensuremath{\\psi(2\\mathrm{S})}$ (top right), \n and the prompt (bottom left) and non-prompt (bottom right) ratios for 7 \\TeV, \n shown for the region $0.75<|y|<1.00$.}\n \\label{fig:fracsyst_ratio}\n \\end{center}\n\\end{figure}\n\n\nThe sources of systematic uncertainties that are applied to the $\\psi$ double differential cross-section\nmeasurements are from uncertainties in: the luminosity determination; muon and trigger efficiency corrections; inner detector tracking efficiencies; the fit model parametrization; and due to bin migration corrections.\nFor the non-prompt fraction and ratio measurements the systematic uncertainties are assessed in the same \nmanner as for the uncertainties on the cross-section, \nexcept that in these ratios some systematic uncertainties, such as the luminosity uncertainty, cancel out.\nThe sources of systematic uncertainty evaluated for the prompt and non-prompt $\\ensuremath{\\psi}$ cross-section measurements, along with the minimum, maximum and median values, are listed in Table~\\ref{table:systlist}.\nThe largest contributions, which originate from the trigger and fit model uncertainties, are typically for the \nhigh $\\mbox{$p_{\\text{T}}$}$ intervals and are due to the limited statistics of the efficiency maps (for the trigger), and the data sample (for the fit model).\n\nFigures~\\ref{fig:fracsyst_xs} and~\\ref{fig:fracsyst_ratio} show, for a representative interval,\nthe impact of the considered uncertainties\non the production cross-section, as well as the non-prompt fraction and ratios for $7$~\\TeV\\ data.\nThe impact is very similar at 8 \\TeV.\n\n\n\\begin{table}[h!]\n \\caption{Summary of the minimum and maximum contributions along with the median value of the systematic uncertainties as percentages for the prompt and non-prompt $\\psi$ cross-section results.\n Values are quoted for 7 and 8 \\TeV\\ data.}\n \\centering\n \\begin{tabular}{l || c c c | c c c }\n \\hline \n & \\multicolumn{3}{c|}{7 \\TeV\\ [\\%]} & \\multicolumn{3}{c}{8 \\TeV\\ [\\%]} \\\\\n \\hline \n Source of systematic uncertainty & Min & Median & Max & Min & Median & Max \\\\\n Luminosity & 1.8 & 1.8 & 1.8 & 2.8 & 2.8 & 2.8 \\\\\n Muon reconstruction efficiency & 0.7 & 1.2 & 4.7 & 0.3 & 0.7 & 6.0 \\\\\n Muon trigger efficiency & 3.2 & 4.7 & 35.9 & 2.9 & 7.0 & 23.4 \\\\\n Inner detector tracking efficiency & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\\\\n Fit model parameterizations & 0.5 & 2.2 & 22.6 & 0.26 & 1.07 & 24.9 \\\\\n Bin migrations & 0.01 & 0.1 & 1.4 & 0.01 & 0.3 & 1.5 \\\\ \\hline\n Total & 4.2 & 6.5 & 36.3 & 4.4 & 8.1 & 27.9 \\\\\n \\hline \n \\hline\n \\end{tabular}\n\\label{table:systlist}\n\\end{table}\n\n\n\\setdescription{font=\\normalfont\\itshape}\n\n\\begin{description}[style=unboxed,leftmargin=0cm]\n\\item[Luminosity] \\hfill \\\\\nThe uncertainty on the integrated luminosity is $1.8\\%$ ($2.8\\%$) for the 7~\\TeV\\ (8~\\TeV) data-taking period. \nThe methodology used to determine these uncertainties is described in Ref.~\\cite{Aad:2013ucp}.\nThe luminosity uncertainty is only applied to the \\jpsi\\ and \\ensuremath{\\psi(2\\mathrm{S})}\\ cross-section results.\n\n\n\\item[Muon reconstruction and trigger efficiencies] \\hfill \\\\\nTo determine the systematic uncertainty on the muon reconstruction and trigger efficiency maps, each of the maps is reproduced in 100 pseudo-experiments.\nThe dominant uncertainty in each bin is statistical and hence any bin-to-bin correlations are neglected.\nFor each pseudo-experiment a new map is created by varying independently each bin content according to a Gaussian distribution about its estimated value, \ndetermined from the original map.\nIn each pseudo-experiment, the total weight is recalculated for each dimuon $\\mbox{$p_{\\text{T}}$}$ and $|y|$ interval of the analysis.\nThe RMS of the total weight pseudo-experiment distributions for each efficiency type is used as the systematic uncertainty,\nwhere any correlation effects between the muon and trigger efficiencies can be neglected.\n\n\nThe ID tracking efficiency is in excess of $99.5\\%$ \\cite{Aad2012dlq}, and\nan uncertainty of 1\\% is applied to account for the ID dimuon reconstruction inefficiency (0.5\\% per muon, added coherently). This\nuncertainty is applied to the differential cross-sections and is assumed to cancel in the fraction of non-prompt to inclusive production for \\jpsi\\ and \n\\ensuremath{\\psi(2\\mathrm{S})}\\ and in the ratios of \\ensuremath{\\psi(2\\mathrm{S})}\\ to \\jpsi\\ production.\n\nFor the trigger efficiency $\\epsilon_\\mathrm{trig}$, in addition to the trigger efficiency map, \nthere is an additional correction term that accounts for inefficiencies due to correlations between the two trigger muons, such as the dimuon opening angle.\nThis correction is varied by its uncertainty, and the shift in the resultant total weight relative to its central value is added in quadrature to the uncertainty from the map.\nThe choice of triggers is known \\cite{Aad:2014cqa} to introduce a small lifetime-dependent efficiency loss but it is determined to have a negligible effect on the prompt and non-prompt yields and no correction is applied in this analysis.\nSimilarly, the muon reconstruction efficiency corrections of prompt and non-prompt signals are found to be consistent within the statistical uncertainties of the efficiency measurements, and no additional uncertainty is applied.\n\n\n \\item[Fit model uncertainty] \\hfill \\\\\nThe uncertainty due to the fit procedure is determined by varying one component at a\ntime in the fit model described in Section \\ref{sec:method:fit}, creating a set of new fit models. For each new\nfit model, all measured quantities are recalculated, and in each $\\mbox{$p_{\\text{T}}$}$ and $|y|$ interval the spread of\nvariations around the central fit model is used as its systematic uncertainty.\nThe variations of the fit model also account for possible uncertainties due to final-state radiation.\nThe following variations to the central model fit are evaluated: \n\n\\begin{itemize}\n\n \\item signal mass model --- using double Gaussian models in place of the Crystal Ball plus Gaussian model; variation of the\n $\\alpha$ and $n$ parameters of the $B$ model, which are originally fixed;\n\n \\item signal pseudo-proper decay time model --- a double exponential function is used to describe the pseudo-proper decay time distribution for the \n $\\ensuremath{\\psi}$ non-prompt signal;\n\n \\item background mass models --- variations of the mass model using exponentials functions, or quadratic Chebyshev polynomials to describe the components of prompt,\n non-prompt and double-sided background terms;\n\n \\item background pseudo-proper decay time model --- a single exponential function was considered for the non-prompt component;\n\n \\item pseudo-proper decay time resolution model --- using a single Gaussian function in place of the double Gaussian function to model the lifetime resolution\n (also prompt lifetime model); and variation of the mixing terms for the two Gaussian components of this term.\n\n\\end{itemize}\n\nOf the variations considered, it is typically the parametrizations of the signal mass model and pseudo-proper decay time resolution model that dominate the contribution to the fit model uncertainty.\n\n \\item[Bin migrations] \\hfill \\\\\nAs the corrections to the results due to bin migration effects are factors close to unity in all regions, the difference between the correction factor and unity is applied as the uncertainty. \n\n\\end{description}\n\nThe variation of the acceptance corrections with spin-alignment is treated\nseparately, and scaling factors supplied in Appendix~\\ref{sec:spincorrection}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThermal contact between two macroscopic bodies initially at different temperatures corresponds to a situation where the heat transfer between the two bodies is ensured by a thin diathermal interface. The latter may be an immaterial interface between two solids or a diathermal wall between two fluids. Over a time window during which the two macroscopic bodies have negligible energy variations, they behave as thermostats with constant thermodynamic temperatures, while the interface is a mesoscopic system with traceable configurations. After a long enough time inside the considered time window the interface tends to a stationary non equilibrium state where the instantaneous heat current which it receives from a thermostat has a non-vanishing mean value. Even if the interface is described by a model where its degrees of freedom obey a (deterministic or stochastic) microscopic dynamics, there is no general framework, such as Gibbs equilibrium ensemble theory, which would allow to determine the probability distribution of the interface configurations and the corresponding mean instantaneous heat current. Therefore it is most valuable to exhibit solvable models which would shed some light into the dependence of the heat instantaneous current upon the model parameters and the temperatures of the two thermostats.\n\n\nSuch a solvable model has been introduced by Racz and Zia in 1994 \\cite{RaczZia1994}. They consider a chain of classical spins with periodic boundary conditions and interacting with a nearest-neighbor ferromagnetic (Ising) interaction. They endow it with a Glauber stochastic dynamics with single-spin flips at a time and such that the spins on odd and even lattice sites are flipped by thermostats at two different temperatures. In our view, the model may be seen as a zig-zag shaped chain inside a very thin strip between two half-planes occupied by two different thermostats, and where the odd (even) sites are located on the left (right) side of the strip. The stationary two-spin correlations at any distance as well as higher order spin correlations have been extensively studied in Refs.\\cite{RaczZia1994,SchmittmannSchmuser2002,MobiliaZiaSchmittmann2004,MobiliaSchmittmannZia2005}.\n\n\n\nWe point out that the latter model indeed satisfies two requirements needed for a correct description of a thermal contact between two macroscopic bodies during a transient time window where their temperature can be considered as constant. First the contact must be mediated by changes in the internal energy of the spin interface. Second, if the energies of the macroscopic bodies were kept tracked of, the transition rates for both the interface configurations and these two energies would obey the detailed balance with the microcanonical equilibrium probability distribution for these variables of the whole system. Then, in the infinite time limit the two bodies and the interface would be at the same temperature; in a time window where the energy variations of the macroscopic bodies are negligible, the transition rates for the interface depend only on the temperatures of the macroscopic bodies and they obey the local detailed balance \\cite{Derrida2007,BauerCornu2015}\\footnote{In Ref.\\cite{Derrida2007} the terminology \"generalized\" detailed balance is used.}. \nIn the case of a spin interface the two corresponding requirements become: 1) a transition between two spin configurations involves only one thermostat; 2) the transition rate involving a given thermostat obey the detailed balance at the same temperature. For the present two-temperature Ising chain the first requirement is obviously fulfilled\\footnote{We notice that the first requirement is not satisfied by the Ising chain model of Ref.\\cite{CornuHilhorst2017}, where two thermostats at different temperatures act on every spin : the latter model does not describe a situation of thermal contact.}, and the\nsimplest transition rates which fulfill the second requirement are those chosen by Glauber in the case of an Ising chain in contact with a unique thermostat\\cite{Glauber1963}. (We recall that Glauber looked for single-spin flip dynamics such that in the infinite time limit the spin chain indeed relaxes to the canonical equilibrium state at the thermostat temperature and then he chose the simplest transition rates.)\n\n\\medskip\n\nOn the other hand, the quantities of interest for exchange processes which have been considered over the last three decades, both theoretically and experimentally, are amounts of microscopically conserved entities (matter, energy, ...), which are exchanged over a very long time (in the case of thermal contact the corresponding quantities are the heat amounts received by the interface from each thermostat during a fixed long time interval). They have been focused on because the scaled generating function for their cumulants per unit of time in the long time limit as well as its Laplace transform, the large deviation function of the corresponding time-integrated current, have been shown to obey generic symmetry relations, the so-called fluctuation relations. The latter relations are derived from properties of the system dynamics and they are a milestone of \nthe stochastic thermodynamics theory (For a review see Ref.\\cite{Seifert2012}.) \n\n\nIn this context it is also interesting to have at hand solvable models where the statistics of the time-integrated currents can be calculated. For instance, such a model has been exhibited in the case where the heat transfer between two thermostats is ensured by a wire. The energy quanta are represented as particles whose stochastic dynamics is a Symmetric Simple Exclusion Process (SSEP) with adequate boundary conditions: particles with hard cores hop on the sites of a one-dimensional lattice with the same hopping rate in both directions, but have different in-coming and out-going rates at the two lattice ends in contact with two particle reservoirs with different chemical potentials. Then all cumulants of the heat coming out of one thermostat per unit of time can be calculated in the long time limit \\cite{DerridaDoucotRoche2004,BodineauDerrida2007} .\n\n\nIn the present paper we consider a generalized version of the diathermal interface model of Ref.\\cite{RaczZia1994}: the two Glauber dynamics which flip the spins on sites with odd or even indices respectively have not only different temperatures but also different kinetic parameters (see Sec.\\ref{secModel}). \nOur aim is to calculate the scaled generating function of the joint cumulants per unit of time for the heats flowing out of the two thermostats in the long time limit. \n\nThe difference between the kinetic constants is relevant for two macroscopic bodies made with different materials. Some of the corresponding kinetic effects have been previously investigated in our study of a very simple model for a one-dimensional interface between two half-planes occupied by bodies at two different temperatures : a model of independent two-spin pairs where the left-side (right-side) spin in each pair is flipped only by the thermostat on the same side according to a Glauber dynamics. The whole statistics for the spin configurations and the heat amounts exchanged by every pair with the thermostats have been calculated explicitly \\cite{CornuBauer2013}.\n\n\n\n \n\\medskip\nThe method which we use to obtain the scaled generating function for the heat cumulants in the present model is the following.\nFrom its definition this function can be obtained as the largest eigenvalue of the modified Markov matrix which rules the evolution of the joint probability for the system configurations and \n the heat amounts $\\Heat_\\text{\\mdseries o}$ and $\\Heat_\\text{\\mdseries e}$ received on each (odd or even) sublattice. \n \nIn order to study energy exchanges more conveniently, instead of considering the spin configurations, we rather formulate the problem in terms of the position configurations for the domain walls, which sit on the dual lattice. In other words we consider the well-known lattice gas representation on the {\\it dual\\,} lattice (`antiparallel adjacent spin pair' $\\leftrightarrow$ `particle' and `parallel adjacent spin pair' $\\leftrightarrow$ `hole') where a particle is in fact a domain wall. The mapping of Glauber spin dynamics to the particle dynamics then includes the possibility for the creation and annihilation of adjacent particle pairs, which correspond to the injection or the loss of energy in the chain respectively\\footnote{The latter correspondance has been used for instance in Ref.\\cite{FaragoPitard2007} for the calculation of the scaled generating function of the energy injected in an Ising spin ring through the random flips of one spin while all other spins evolve according to a Glauber dynamics at zero temperature which dissipates energy along the ring.}. Our model with two temperatures and two kinetic parameters is mapped to a reaction-diffusion system with two different creation (annihilation) rates as well as two different hopping rates on spatially alternating sites\\footnote{In Ref.\\cite{MobiliaSchmittmannZia2005} the mapping has been used in the reverse sense in order to study the relaxation towards the stationary state for the reaction-diffusion system from results obtained for the Ising spin chain dynamics with two temperatures but a unique kinetic parameter.}.\n\n\n\nThe crucial point is that, in a suitable basis for the representation of the configurations of domain wall positions, the Markov matrix for the evolution of the configuration probability involves only products of two operators : the Markov matrix is mapped to a free fermion Hamiltonian \\cite{Schutz2001}.\nMoreover the mapping can be readily generalized for the modified Markov matrix which rules the evolution of the joint probability for a configuration of domain wall positions and the amounts of heat $\\Heat_\\text{\\mdseries o}$ and $\\Heat_\\text{\\mdseries e}$.\nThe latter matrix can be diagonalized by using free fermions techniques: \n Jordan-Wigner transformation and antiperiodic Fourier transform. Thus we obtain a block diagonal matrix, made of $4\\times 4$ blocks, which can be straighforwardly diagonalized by introducing four pseudo-fermion operators. \n (In the case of two pseudo-fermion operators and the associated Bogoliubov-like transformation, see for instance Ref.\n\\cite{GrynbergStinchcombe1995PRL,GrynbergStinchcombe1995PRE}.)\n The largest eigenvalue of the modified Markov matrix is obtained by filling up all pseudo-fermion states associated with an eigenvalue with a positive real part. \n\n \n\n\\medskip\n\n The paper is organized as follows. The model is defined in Sec.\\ref{secModel} and the relaxation time for the mean global magnetization on each sublattice is calculated. Moreover by solving a hierarchy of equations for the stationary global two-spin correlations at any distance (see Appendix \\ref{secTwospincorrelations}), we determine the mean instantaneous energy current which flows from each thermostat into the spin lattice in the stationary state. The mapping to the dynamics of domain walls on the dual lattice and the associated modified matrix is derived in Sec.\\ref{SecMapping}. Its eigenvalues are determined in Sec.\\ref{SecEigenvalues} by free fermion techniques. The cumulants are obtained in Sec.\\ref{SecCumulants} and various physical regimes are discussed. In conclusion we summarize some finite-size effects and their possible cancellation in heat cumulants. \n\n \n\n \n\\section{Model}\n\\label{secModel}\n\n\\subsection{Description of a thermalization process}\n\nWe consider a one-dimensional lattice with a finite even number of sites $L=2N$, where each site $j$ is occupied by a classical spin $s_j$ ($s_j=\\pm 1$, $j=1,\\ldots, 2N$) with periodic boundary condition $s_{j+2N}=s_j$. Spins interact via the Ising ferromagnetic nearest-neighbor interaction with coupling $K>0$: the energy of a spin configuration $\\mathbf{s}$ is\n\\begin{equation}\n\\label{defEnergie}\n{\\cal E}(\\mathbf{s})=-K \\sum_{j=1}^{2N}s_{j}s_{j+1}.\n\\end{equation}\nWhen the spin at site $j$ is flipped, the energy variation of the Ising chain is equal to\n\\begin{equation}\n\\label{varEnergie}\n\\Delta {\\cal E}(s_j\\to -s_j)= s_j\\frac{s_{j-1}+s_{j+1}}{2} \\Delta E,\n\\end{equation}\nwith $\\Delta E=4K$. This variation can take the values $+\\Delta E$, 0, or $-\\Delta E$, \n\n\nThe model is endowed with a stochastic dynamics where the spin flips at odd (even) sites are due to energy exchanges with a macroscopic body at temperature $T_\\text{\\mdseries o}$ ($T_\\text{\\mdseries e}$) in the course of a thermalization process of the two macroscopic bodies. The transition rates must obey local detailed balance \\cite{Derrida2007,BauerCornu2015}: \nthe transition rates $w(s_j \\to -s_j)$ and $w(-s_j \\to s_j)$ for two reversed flips of the spin at site $j$, while all other spins are kept fixed, must obey the ratio\n\\begin{equation}\n\\label{LDB}\n\\frac{w(s_j \\to -s_j)}{w(-s_j \\to s_j)}=\\text{\\mdseries e}^{-\\beta_j \\Delta {\\cal E}(s_j\\to -s_j)},\n\\end{equation}\nwhere $\\beta_j$ is the inverse temperature of the thermostat acting on site $j$ : $\\beta_j=1\/(k_{\\scriptscriptstyle B} T_j)$ where $k_{\\scriptscriptstyle B}$ is Boltzmann constant and $T_j$ is equal either to $T_\\text{\\mdseries o}$ or $T_\\text{\\mdseries e}$, depending on the parity of $j$.\nAs shown by Glauber \\cite{Glauber1963} in the case of a unique temperature, the simplest transition rates which obey the constraint \\eqref{LDB} read\n\\begin{equation}\n\\label{deftauxtrans}\nw(s_j \\to -s_j)=\\frac{\\nu_j}{2} \\left[1-\\gamma_j \\frac{s_j \\left(s_{j-1}+s_{j+1}\\right)}{2}\\right],\n\\end{equation}\nwhile $w(-s_j \\to s_j)$ is given by the latter expression where $s_j$ is replaced by $-s_j$.\nIn \\eqref{deftauxtrans} $\\gamma_j $ is the thermodynamic parameter at site $j$,\n\\begin{equation}\n\\label{defgammaa}\n\\gamma_j =\\tanh\\left(\\frac{\\beta_j \\Delta E}{2}\\right)\n\\end{equation}\nwhere $\\gamma_j=\\gamma_\\text{\\mdseries o}$ or $\\gamma_\\text{\\mdseries e}$, depending on the parity of the site index,\nand $\\nu_j$ is the kinetic parameter $\\nu_j$ at site $j$. The latter is not determined by the local detailed balance; it can be interpreted as the mean frequency at which the macroscopic body tries to flip the spin at site $j$. Therefore we set $\\nu_j=\\nu_\\text{\\mdseries o}$ or $\\nu_\\text{\\mdseries e}$, depending on the parity of the site index (note that when the two kinetic parameters are equal the present model coincides with that of Ref.\\cite{RaczZia1994}).\nSince the time scale is arbitrary, it is convenient to introduce the dimensionless kinetic parameters $\\overline{\\nu}_\\text{\\mdseries a}$, with $\\text{a}=\\text{o}$ or $\\text{e}$, defined as\n\\begin{equation}\n\\label{defDimensionlessNu}\n\\overline{\\nu}_{\\text a}= \\frac{\\nu_{\\text a}}{\\nu_\\text{\\mdseries o} +\\nu_\\text{\\mdseries e}}.\n\\end{equation}\nThey satisfy the relation $\\overline{\\nu}_\\text{\\mdseries o} +\\overline{\\nu}_\\text{\\mdseries e}=1$ and, apart from the arbitrary time scale, the model has only three independent parameters, $\\gamma_\\text{\\mdseries o}$, $\\gamma_\\text{\\mdseries e}$ and $\\overline{\\nu}_\\text{\\mdseries o}$.\n\n\n\nThe probability $P(\\mathbf{s};t)$ for the system to be in spin configuration $\\mathbf{s}$ at time $t$ evolves according to the master equation\n\\begin{equation}\n\\label{MarkovSpins}\n\\frac{\\text{\\mdseries d} P(\\mathbf{s};t)}{\\text{\\mdseries d} t}=\\sum_j^{2N}w(-s_j \\to s_j)P(\\mathbf{s}_j;t)- \\left(\\sum_j^{2N} w(s_j \\to -s_j) \\right) P(\\mathbf{s};t)\n\\end{equation}\nwhere $\\mathbf{s}_j$ denotes the spin configuration obtained from $\\mathbf{s}$ by changing $s_j$ into $-s_j$.\n The number of configurations is finite, and the transition rates allow the system to evolve from any configuration to any other one after a suitable succession of transitions. Therefore there is a unique stationary solution of the master equation. In the following we focus on the mean values of global quantities and denote $\\Esp{\\cdots}$ and $\\Espst{\\cdots}$ the expectation values calculated with the time-dependent probability $P(\\mathbf{s};t)$ and the stationary probability $P_\\text{\\mdseries st}(\\mathbf{s})$ respectively.\n\n\\subsection{Relaxation of the mean global sublattice magnetizations}\n\\label{secMagnetization}\n\n\n The transition rates are invariant under a global flip of the spins, so that\n a configuration and the corresponding one where all spins are flipped have the same probability in the stationary state. As a result all stationary correlations for an odd number of spins vanish identically; in particular $\\Espst{s_j}=0$. As a consequence the mean values of the global magnetizations on the two sublattices, $M_\\text{\\mdseries o}=\\sum_{n=1}^{N}s_{2n-1}$ and $M_\\text{\\mdseries e}=\\sum_{n=1}^{N} s_{2n}$ respectively, vanish in the stationary state,\n \\begin{equation}\n \\Espst{M_\\text{\\mdseries o}}= \\Espst{M_\\text{\\mdseries e}}=0.\n \\end{equation}\n \nThe relaxation of the mean values of global sublattice magnetizations is readily studied.\nAs in the case of the homogeneous spin chain considered by Glauber \\cite{Glauber1963}, the evolution equation for the mean value of the spin at site $j$ reads\n\\begin{equation}\n\\frac{\\text{\\mdseries d} \\Esp{s_j}}{\\text{\\mdseries d} t} = - 2 \\Esp{s_j w(s_j \\to -s_j)}.\n\\end{equation}\nAccording to the expression of the transition rates \\eqref{deftauxtrans}\n\\begin{equation}\n\\frac{\\text{\\mdseries d} \\Esp{s_j}}{\\text{\\mdseries d} t} = -\\nu_j \\left[\\Esp{s_j} - \\gamma_j \\frac{\\Esp{s_{j-1}}+\\Esp{s_{j+1}}}{2} \\right].\n\\end{equation}\nThen the coupled evolutions of the magnetizations on the two sublattices read\n\\begin{eqnarray}\n\\frac{\\text{\\mdseries d} \\Esp{M_\\text{\\mdseries o}}}{\\text{\\mdseries d} t} &=& - \\nu_\\text{\\mdseries o}\\left[ \\Esp{M_\\text{\\mdseries o}} - \\gamma_\\text{\\mdseries o} \\Esp{M_\\text{\\mdseries e}}\\right]\n\\\\ \\nonumber\n\\frac{\\text{\\mdseries d} \\Esp{M_\\text{\\mdseries e}}}{\\text{\\mdseries d} t} &=& - \\nu_\\text{\\mdseries e}\\left[ \\Esp{M_\\text{\\mdseries e}} - \\gamma_\\text{\\mdseries e} \\Esp{M_\\text{\\mdseries o}}\\right].\n\\end{eqnarray}\nFrom these equations we retrieve that both mean magnetizations vanish in the stationary state, as predicted by symmetry arguments. The matrix associated with this system of linear equations has two strictly negative eigenvalues \n$\\tfrac{1}{2}(\\nu_\\text{\\mdseries o}+\\nu_\\text{\\mdseries e})\\left[-1\\pm\\sqrt{(\\overline{\\nu}_\\text{\\mdseries o}-\\overline{\\nu}_\\text{\\mdseries e})^2+4 \\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e}\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}}\\right]$, each of which is associated with a couple of right and left eigenvectors (the eigenvalues are negative because\n$(\\overline{\\nu}_\\text{\\mdseries o}-\\overline{\\nu}_\\text{\\mdseries e})^2+4 \\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e}\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}=1-4 \\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e}(1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e})<1$). For generic values of the initial magnetizations $M_\\text{\\mdseries o}$ and $M_\\text{\\mdseries e}$, the inverse relaxation time $1\/t_\\text{\\mdseries rel}$ to their stationary value is given by the opposite of the negative eigenvalue with the smallest modulus, and the relaxation time $t_\\text{\\mdseries rel}$ reads\n\\begin{equation}\n\\label{trel}\nt_\\text{\\mdseries rel}=\\frac{2}{\\nu_\\text{\\mdseries o}+\\nu_\\text{\\mdseries e}} \\left[ 1-\\sqrt{(\\overline{\\nu}_\\text{\\mdseries o}-\\overline{\\nu}_\\text{\\mdseries e})^2+4 \\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e}\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}}\\right]^{-1}.\n\\end{equation}\n\n\n\n\n\n\\subsection{Mean global heat current in the stationary state}\n\nThe mean instantaneous heat current $\\Esp{j_k}$ received by the spin chain at site $k$ from the thermostat at temperature $T_k$ is equal to the expectation value of the variation of the chain energy when the spin $s_k$ is flipped times the transition rate for the flip. According to the expressions for the energy variation \\eqref{varEnergie}\nand for the transition rates \\eqref{deftauxtrans}, the mean instantaneous current reads\n\\begin{equation}\n\\Esp{j_k}=K\\nu_k\\left[ -\\gamma_k -\\gamma_k \\Esp{s_{k-1}s_{k+1}} + \\Esp{s_{k-1} s_k} +\\Esp{s_k s_{k+1}}\\right].\n\\end{equation}\nTherefore the stationary mean value of the global heat current coming from the thermostat acting on spins at even sites, namely\n$\nJ_\\text{\\mdseries e}= \\sum_{n=1}^N j_{2n},\n$\nis determined as\n\\begin{equation}\n\\label{ExpressionJe}\n\\Espst{J_\\text{\\mdseries e}}= N K \\nu_\\text{\\mdseries e} \\left[ -\\gamma_\\text{\\mdseries e} -\\gamma_\\text{\\mdseries e} D^\\text{oo}_2 + D^\\text{oe}_1 + D^\\text{eo}_1\\right]\n\\end{equation}\nwith the following definitions : $D^\\text{oo}_2$ is the average over the sublattice of odd sites of the stationary correlation between two spins separated by two sites,\n\\begin{equation}\nD^\\text{oo}_2=\\frac{1}{N}\\sum_{n=1}^N \\Espst{s_{2n-1} s_{2n+1}},\n\\end{equation} \n$D^\\text{oe}_1=(1\/N)\\sum_{n=1}^N \\Espst{s_{2n-1} s_{2n}}$ \nand $D^\\text{eo}_1$ has an analogous definition. Similarly the stationary mean value of the global heat current coming from the thermostat acting on spins at odd sites, $J_\\text{\\mdseries o}= \\sum_{n=1}^N j_{2n-1}$, reads\n\\begin{equation}\n\\label{ExpressionJo}\n\\Espst{J_\\text{\\mdseries o}}= N K \\nu_\\text{\\mdseries o} \\left[ -\\gamma_\\text{\\mdseries o} -\\gamma_\\text{\\mdseries o} D^\\text{ee}_2 + D^\\text{oe}_1 + D^\\text{eo}_1\\right]\n\\end{equation}\nwith $D^\\text{ee}_{2}=(1\/N)\\sum_{n=1}^N \\Espst{s_{2n} s_{2n+2}}$.\n\n\n\n\\medskip\nThe values of the stationary global two-spin correlations $D^\\text{oo}_2$, $D^\\text{ee}_2$, $D^\\text{oe}_1$ and $D^\\text{eo}_1$ can be determined from a hierarchy of equations for similar quantities with two spins at any distance on the lattice. Details are given in Appendix \\ref{secTwospincorrelations} with the results\n \\eqref{relDooDee}, \\eqref{relDoeDeo}, \\eqref{valueDee}, and \\eqref{valueDoe} for any distance between spins. From the latter results we get\n\\begin{equation}\n\\label{valueDee2}\nD^\\text{ee}_2=\\frac{\\gamma \\eta_-}{\\gamma_\\text{\\mdseries o}} \\, \\frac{1 +\\eta_-^{N- 2} }{1+\\eta_-^N} \n\\end{equation}\nwith\n\\begin{equation}\n\\label{defgamma}\n\\gamma= \\overline{\\nu}_\\text{\\mdseries o} \\gamma_\\text{\\mdseries o} + \\overline{\\nu}_\\text{\\mdseries e} \\gamma_\\text{\\mdseries e},\n\\end{equation}\nwhere the dimensionless kinetic parameters have been defined in \\eqref{defDimensionlessNu}, and $\\eta_-$, with $0< \\eta_- <1$, is defined by\n\\begin{equation}\n\\sqrt{\\eta_-}=\\frac{1-\\sqrt{1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}}}{\\sqrt{\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}}}.\n\\end{equation}\nMoreover $D^\\text{oo}_2=(\\gamma_\\text{\\mdseries o}\/\\gamma_\\text{\\mdseries e}) D^\\text{ee}_2$, while\n\\begin{equation}\n\\label{valueDoe1}\nD^\\text{oe}_1=D^\\text{eo}_1=\\frac{\\gamma \\sqrt{\\eta_-}}{\\sqrt{\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}}} \\,\\frac{1 +\\eta_-^{N-1}}{1+\\eta_-^N}.\n\\end{equation}\nWe notice that the dependence upon the kinetic parameters $\\nu_\\text{\\mdseries o}$ and $\\nu_\\text{\\mdseries e}$ occurs only through the parameter $\\gamma$ defined in \\eqref{defgamma}.\n\nEventually the stationary mean value of the global heat current received on even sites can be calculated from expression \\eqref{ExpressionJe} and relation \\eqref{RelRoots};\nwe get \n\\begin{equation}\n\\label{valueJe}\n\\Espst{J_\\text{\\mdseries e}}= N K \\frac{\\nu_\\text{\\mdseries o} \\nu_\\text{\\mdseries e}}{\\nu_\\text{\\mdseries o}+\\nu_\\text{\\mdseries e}} (\\gamma_\\text{\\mdseries o}-\\gamma_\\text{\\mdseries e}).\n\\end{equation}\nSimilarly the stationary mean value of the global heat current received on odd sites can be obtained from the expression \\eqref{ExpressionJo} ; it proves to be opposite to that on even sites, \n$\\Espst{J_\\text{\\mdseries o}}=-\\Espst{J_\\text{\\mdseries e}}$, as it should in the stationary state where the mean energy of the chain is constant.\n\nWe point out the following remarkable property : though $D^\\text{ee}_2$, $D^\\text{oo}_2$, $D^\\text{oe}_1$ and $D^\\text{eo}_1$ involve finite size corrections (see \\eqref{valueDee2} and \\eqref{valueDoe1}), these corrections cancel one another in the value of the mean global current $\\Espst{J_\\text{\\mdseries e}}$ : $\\Espst{J_\\text{\\mdseries e}}$ is exactly proportional to the size $L=2N$ of the ring. Moreover it happens to be equal to $L$ times the mean current received by a spin in the independent pair model of Ref.\\cite{CornuBauer2013}.\n\n\n\n\n\\section{Mapping to a reaction-diffusion system with pair creation-annihilation}\n\\label{SecMapping}\n\nTo prepare the study of the heat amounts exchanged with the thermostats we consider a mapping to another model for which the evolution operator is quadratic. \n\n\\subsection{Domain wall system}\n\nWhen two spins on neighboring sites are antiparallel, one may consider that there is a domain wall between them, whereas there is no domain wall when they are parallel. The domain walls sit on the edges of the initial lattice. Labeling each edge by its mid-point, one gets another lattice which we call the dual lattice in what follows. The edge $(j-1,j)$ and the corresponding site on the dual lattice are labeled by $j$.\n If $s_{j-1}$ and $s_j$ are antiparallel, $s_{j-1}s_j=-1$, then the occupation number by a domain wall at site $j$ on the dual lattice is $n_j=1$, whereas if $s_{j-1}$ and $s_j$ are parallel $n_j=0$. Thus the correspondance reads\n\\begin{equation}\n\\label{CorrespondenceWallSpin}\nn_j=\\frac{1-s_{j-1}s_{j}}{2}.\n\\end{equation}\nOn a ring the number of domain walls is even and $\\sum_{j=1}^L n_j$ is even.\n\nAs a result a spin configuration can be characterized either by the set $\\mathbf{s}=\\{s_1,\\cdots s_L\\}$ of spin configurations or by the knowledge of the value of $s_1$ and the set of the positions of the domains walls, \nnamely the set of occupations numbers $\\mathbf{n}=\\{n_1,\\cdots,n_L\\}$. \nThe energy of the system can be expressed solely in terms of domain walls as\n\\begin{equation}\n{\\cal E}(\\mathbf{n})= - 2N K+2 K \\sum_{j=1}^{2N} n_j.\n\\end{equation}\n\n\n\n\\subsection{Quantum mechanics notations}\n\nIn the following we use the quantum mechanics notations, as commonly done in the literature. Then a column vector is denoted as a ``ket'', $\\ket{\\ldots}$ and a row vector is denoted as a ``bra'' $\\bra{\\ldots}$.\nThe configuration of occupation numbers by domain walls ${\\mathbf{n}}=\\{n_1, \\cdots, n_{L}\\}$ is represented as a tensor product\n\\begin{equation}\n\\label{tensorialprod}\n \\ket{\\mathbf{n}}=\\otimes_{j=1}^{L} \\ket{n_j},\n\\end{equation}\nwhere $\\ket{n_j}$ is a two-component column vector.\nThe convention used for kets associated to vacant and occupied states is \n\\begin{equation}\n\\label{convention-vacant}\n\\veccolumn{1}{0}_j=\\ket{n_j=0}\\quad\\textrm{and}\\quad \\veccolumn{0}{1}_j=\\ket{n_j=1}.\n\\end{equation}\nThis convention is the standard choice of basis in the condensed matter literature on quantum spin chains.\nWith the representation \\eqref{tensorialprod}-\\eqref{convention-vacant} the row-column product $\\brat{\\mathbf{n'}}\\ket{\\mathbf{n}}$ takes the form\n$\n\\brat{\\mathbf{n'}}\\ket{\\mathbf{n}}=\\prod_{j=1}^{L} \\delta_{n'_j,n_j}\n$.\nTherefore the probability of the domain wall configuration $\\mathbf{n}$ at time $t$, $P(\\mathbf{n};t)$,\ncan be represented as a row-column (scalar) product \n$\nP(\\mathbf{n};t)= \\brat{\\mathbf{n}} \\ket{P_t}\n$,\nwhere $\\ket{P_t}$ is the column vector defined as\n\\begin{equation}\n\\label{defketProb}\n\\ket{P_t}=\\sum_{\\mathbf{n}} P(\\mathbf{n};t) \\ket{\\mathbf{n}}.\n\\end{equation}\n\n\nWith the latter definitions the master equation for the stochastic evolution of the probability $P(\\mathbf{n};t)$, which takes the generic form written in \\eqref{MarkovSpins} in the case of $P(\\mathbf{s};t)$, can be represented as the evolution of the column vector $\\ket{P_t}$ under the Markov matrix $\\mathbb{M}$\n\\begin{equation}\n\\frac{\\text{\\mdseries d} \\ket{P_t}}{\\text{\\mdseries d} t}=\\mathbb{M} \\ket{P_t}\n\\end{equation}\nwith \n\\begin{eqnarray}\n\\bra{\\mathbf{n}'} \\mathbb{M} \\ket{\\mathbf{n}}&=& w(\\mathbf{n} \\to \\mathbf{n}')\n \\quad\\textrm{if}\\quad \\mathbf{n}'\\neq \\mathbf{n}\n\\\\ \\nonumber\n\\bra{\\mathbf{n}} \\mathbb{M} \\ket{\\mathbf{n}}&=&-\\sum_{\\mathbf{n}'\\neq\\mathbf{n}} w(\\mathbf{n} \\to \\mathbf{n}'),\n\\end{eqnarray}\nwhere $w(\\mathbf{n} \\to \\mathbf{n}')$ denotes the transition rate from configuration $\\mathbf{n}$ to configuration $\\mathbf{n}'$.\n\n\n\n\\subsection{Markov matrix for the model}\n\n\nFor the present model of domain walls the matrix elements of $\\mathbb{M}$ can be expressed in terms of Pauli matrices. Indeed the operator for the occupation number at site $j$ reads \n\\begin{equation}\n\\label{widehatn}\n\\widehat{n}_j=\\tfrac{1}{2} \\left( \\mathbb{1}_j-\\sigma^z_j\\right)\n\\end{equation}\nwhere $\\mathbb{1}_j$ denotes the identity $2\\times 2$ matrix at site $j$, and\n{\\small $\\sigma^z_j=\\begin{pmatrix}\n 1 & 0 \\\\\n 0 & -1\n\\end{pmatrix}_j$};\n the operator which changes the occupation number at site $j$ is \n{\\small $\n\\sigma^x_j=\n\\begin{pmatrix}\n 0 & 1 \\\\\n 1 & 0\n\\end{pmatrix}_j\n$}.\nBy inspection of the transition rates for the spin configurations $w(\\mathbf{s} \\to \\mathbf{s}')$ given by \\eqref{deftauxtrans}, every transition rate for the occupation numbers by domain walls, $w(\\mathbf{n} \\to \\mathbf{n}')$, can be written as a matrix element $\\bra{\\mathbf{n}'} \\mathbb{W} \\ket{\\mathbf{n}}$, where \n\n- for a hop of a domain wall from site $j$ to site $j+1$\n\\begin{equation}\n\\mathbb{W} =\\frac{\\nu_j}{2}\\sigma^x_j \\sigma^x_{j+1} \\widehat{n}_j \\left(\\mathbb{1}_j-\\widehat{n}_{j+1}\\right)\n\\end{equation}\n\n- for a hop of a domain wall from site $j+1$ to site $j$\n\\begin{equation}\n\\mathbb{W} =\\frac{\\nu_j}{2}\\sigma^x_j \\sigma^x_{j+1} \\left( \\mathbb{1}_j-\\widehat{n}_j\\right) \\widehat{n}_{j+1}\n\\end{equation}\n\n- for the annihilation of two domain walls at sites $j$ and $j+1$ \n\\begin{equation}\n\\mathbb{W} = \\frac{\\nu_j}{2}(1+\\gamma_j)\\sigma^x_j \\sigma^x_{j+1} \\widehat{n}_j \\widehat{n}_{j+1}\n\\end{equation}\n\n- for the creation of two domain walls at sites $j$ and $j+1$ \n\\begin{equation}\n\\mathbb{W} =\\frac{\\nu_j}{2}(1-\\gamma_j)\\sigma^x_j \\sigma^x_{j+1} \\left( \\mathbb{1}_j-\\widehat{n}_j\\right) \\left( \\mathbb{1}_j-\\widehat{n}_{j+1}\\right).\n\\end{equation}\nThe latter expressions can be written in a more compact form by using the \n spin-$\\tfrac{1}{2}$ ladder operators \n{\\small $\n\\sigma^+=\n\\begin{pmatrix}\n 0 & 1 \\\\\n 0 & 0\n\\end{pmatrix}_j\n$} and \n{\\small $\n\\sigma^-=\n\\begin{pmatrix}\n 0 & 0 \\\\\n 1 & 0\n\\end{pmatrix}_j\n$}. They are such that $\\sigma^x\\widehat{n}=\\sigma^+$ and $\\sigma^x(\\mathbb{1}-\\widehat{n})=\\sigma^-$.\nWith the convention \\eqref{convention-vacant} $\\sigma^+_j$ annihilate a domain wall at site $j$, while $\\sigma^-_j$ creates a domain wall at site $j$.\n\nEventually the Markov matrix $\\mathbb{M}$ derived from the master equation for the evolution of the probability of spin configurations \\eqref{MarkovSpins} reads\n\\begin{eqnarray}\n&&\\mathbb{M} =\\frac{\\nu_1+\\nu_2}{2}\n\\\\ \\nonumber\n&& \\times\n \\left[ -(1-\\gamma) N \\mathbb{1}\n -\\gamma\\sum_{j=1}^{2N} \\widehat{n}_j \n+\\sum_{j=1}^{2N} \\overline{\\nu}_j\\left[ \\sigma^+_j \\sigma^-_{j+1} + \\sigma^-_j \\sigma^+_{j+1}+\n (1+\\gamma_j)\\sigma^+_j \\sigma^+_{j+1} +(1-\\gamma_j)\\sigma^-_j \\sigma^-_{j+1} \\right]\\right],\n\\end{eqnarray}\nwhere $\\mathbb{1}$ denotes the identity $2N \\times 2N$ matrix and $\\gamma$ has been defined in \\eqref{defgamma}. The advantage of the domain wall representation with respect to the spin representation is that the Markov matrix is quadratic in terms of operators acting on different sites instead of involving three operators acting on different sites (for the latter case see for instance Ref.\\cite{Felderhof1971-A,Felderhof1971-B}).\n\n\n\n\\section{Eigenvalues of the modified Markov matrix}\n\\label{SecEigenvalues}\n \n\\subsection{Modified Markov matrix}\n\\label{CumulantsMaximumEigenvalue}\n\nWe are interested in the joint cumulants per unit of time for the heat amounts $\\Heat_\\text{\\mdseries o}$ and $\\Heat_\\text{\\mdseries e}$ which are received by the chain from the thermostat acting on spins at odd and even sites during a time $t$ in the long time limit. The corresponding scaled generating function is\n\\begin{equation}\n\\label{defgL}\ng_{2N}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e};t)= \\lim_{t\\to\\infty} \\frac{1}{t} \\ln\\Esp{ \\text{\\mdseries e}^{\\lambda_\\text{\\mdseries o}\\Heat_\\text{\\mdseries o}+\\lambda_\\text{\\mdseries e}\\Heat_\\text{\\mdseries e}}},\n\\end{equation}\nwhere $\\lambda_\\text{\\mdseries o}$ and $\\lambda_\\text{\\mdseries e}$ are real parameters.\nIn fact an evolution equation can be written for the probability \n$P(\\mathbf{n}, \\Heat_\\text{\\mdseries o},\\Heat_\\text{\\mdseries e};t)$ for the system to be in configuration $\\mathbf{n}$ at time $t$ and to have received heat amounts $\\Heat_\\text{\\mdseries o}$ and $\\Heat_\\text{\\mdseries e}$ between times $0$ and $t$. Therefore \nthe expectation value in the definition \\eqref{defgL} can be expressed in terms of the discrete Laplace transform of \n$P(\\mathbf{n}, \\Heat_\\text{\\mdseries o},\\Heat_\\text{\\mdseries e};t)$, and then $g_{2N}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e};t)$ reads\n\\begin{equation}\n\\label{defgLBis}\ng_{2N}(\\lambda_\\text{\\mdseries e},\\lambda_\\text{\\mdseries o};t)= \\lim_{t\\to\\infty} \\frac{1}{t} \\ln\\sum_{\\mathbf{n}} \\widehat{P}(\\mathbf{n}, \\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e};t)\n\\end{equation}\nwith\n\\begin{equation}\n\\widehat{P}(\\mathbf{n}, \\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e};t)= \\sum_{\\{\\Heat_\\text{\\mdseries o},\\Heat_\\text{\\mdseries e}\\}} \\text{\\mdseries e}^{\\lambda_\\text{\\mdseries o}\\Heat_\\text{\\mdseries o}+\\lambda_\\text{\\mdseries e}\\Heat_\\text{\\mdseries e}}P(\\mathbf{n}, \\Heat_\\text{\\mdseries o},\\Heat_\\text{\\mdseries e};t).\n\\end{equation}\n\n\n By inspection of the transition rates \\eqref{deftauxtrans} and according to the correspondence \\eqref{CorrespondenceWallSpin},\n when the spin at site $j$ is flipped under the action of the thermostat at temperature $T_j$, the variation of \n$Q_j$ is equal to $+\\Delta E$ if a pair of domain walls is created at sites $j$ and $j+1$ , $-\\Delta E$ if a pair of domain walls is annihilated at these sites, $0$ if a domain wall jumps either from $j$ to $j+1$ or from $j+1$ to $j$.\nAs a consequence, with a definition for $ \\ket{\\widehat{P}_t (\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})}$ analogous to that for $\\ket{P_t}$ given in \\eqref{defketProb}, namely $\\brat{\\mathbf{n}}\\ket{\\widehat{P}_t (\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})}=\\widehat{P}(\\mathbf{n}, \\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e};t)$, we get the evolution equation\n\\begin{equation}\n\\label{EvLPProb}\n\\frac{\\text{\\mdseries d} \\ket{\\widehat{P}_t (\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})}}{\\text{\\mdseries d} t}= \\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) \\ket{\\widehat{P}_t(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})},\n\\end{equation}\nwhere the so-called modified Markov matrix $\\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ reads\n\\begin{equation}\n\\label{defMmodified}\n\\frac{2}{\\nu_1+\\nu_2} \\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) =-(1-\\gamma) N \\mathbb{1} -\\gamma\\sum_{j=1}^{2N} \\widehat{n}_j \n+\\sum_{j=1}^{2N} \\overline{\\nu}_j\\left[ \\sigma^+_j \\sigma^-_{j+1} + \\sigma^-_j \\sigma^+_{j+1}+\nb_j\\sigma^+_j \\sigma^+_{j+1} +c_j \\sigma^-_j\\sigma^-_{j+1} \\right].\n\\end{equation}\nThe coefficient $b_j$ ($c_j$) is equal either to $b_\\text{\\mdseries o}$ or $b_\\text{\\mdseries e}$ ($c_\\text{\\mdseries o}$ or $c_\\text{\\mdseries e}$) according to the parity of $j$; with the notation $\\text{a}=\\text{o}$ or $\\text{e}$\n\\begin{equation}\n\\label{defbj}\nb_\\text{\\mdseries a}=(1+\\gamma_\\text{\\mdseries a})\\,\\text{\\mdseries e}^{-\\lambda_\\text{\\mdseries a} \\Delta E}\n\\end{equation}\nand\n\\begin{equation}\n\\label{defcj}\nc_\\text{\\mdseries a}=(1-\\gamma_\\text{\\mdseries a})\\,\\text{\\mdseries e}^{\\lambda_\\text{\\mdseries a}\\Delta E}.\n\\end{equation}\nAccording to the evolution equation \\eqref{EvLPProb} the Laplace transform $\\widehat{P}(\\mathbf{n}, \\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e};t)$ is equal to \\\\ $\\bra{\\mathbf{n}} \\exp[\\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) t] \\ket{\\widehat{P}_{t=0} (\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})}$. \nThus the scaled generating function $g_{2N}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e};t)$ given by \\eqref{defgLBis} is equal to the largest eigenvalue of the matrix $\\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) $ which rules the evolution of $\\ket{\\widehat{P}_t(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})}$.\n\n\n\n\\subsection{Jordan-Wigner transformation}\n\nIn order to find the eigenvalues of the modified Markov matrix $\\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ given by \\eqref{defMmodified} we take advantage of its structure analogous to a free fermion Hamiltonian and we introduce the following Jordan-Wigner transformation \\cite{JordanWigner1928}\n\\begin{equation}\nf^\\dag_j=\\left(\\prod_{k=1}^{j-1}\\sigma^z_k \\right)\\sigma^-_j\n\\quad\\textrm{and}\\quad\nf_j=\\left(\\prod_{k=1}^{j-1}\\sigma^z_k \\right)\\sigma^+_j.\n\\end{equation}\nThe operator $f_j^\\dag$ is indeed the adjoint of $f_j$, because $(\\sigma^z)^\\dag=\\sigma^z$ and $(\\sigma^-)^\\dag=\\sigma^+$.\nOperators $\\sigma$ acting on different sites commute, whereas $\\sigma^z_j$ anticommutes with $\\sigma^+_j$ and $\\sigma^-_j$; moreover $(\\sigma^+_j)^2=0$ and $(\\sigma^-_j)^2=0$. Therefore the operators $f_j$ and $f^\\dag_j$ obey the fermionic anticommutation relations\n\\begin{equation}\n\\{f_j,f_{j'}\\}=0 \\quad \\{f^\\dag_j,f^\\dag_{j'}\\}=0 \\quad \\{f_j,f^\\dag_{j'}\\}=\\delta_{j,j'}.\n\\end{equation}\nThe occupation number of site $j$ by a domain wall, given by \\eqref{widehatn}, also reads $\\widehat{n}_j=\\sigma^-_j\\sigma^+_j=f^\\dag_j f_j$. The expression \\eqref{defMmodified} of the modified matrix $\\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ is rewritten in terms of fermionic operators as\n\\begin{eqnarray}\n\\nonumber\n\\label{widehatM}\n\\frac{2}{\\nu_1+\\nu_2} \\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) \n&=&-(1-\\gamma) N \\mathbb{1} -\\gamma\\sum_{j=1}^{2N}f^\\dag_jf_j\n+\\sum_{j=1}^{2N-1} \\overline{\\nu}_j\\left[f^\\dag_j f_{j+1} - f_jf^\\dag_{j+1} +c_j f^\\dag_jf^\\dag_{j+1} -b_j f_jf_{j+1} \\right] \n\\\\\n&-& \\overline{\\nu}_\\text{\\mdseries e} (-1)^{{\\cal N}_\\text{\\mdseries f}}\\left[ f^\\dag_{2N} f_1 - f_{2N}f^\\dag_1 +c_\\text{\\mdseries e} f^\\dag_{2N}f^\\dag_1-b_\\text{\\mdseries e} f_{2N}f_1\\right]\n\\end{eqnarray}\nwhere ${\\cal N}_\\text{\\mdseries f}=\\sum_{j=1}^{2N} f^\\dag_j f_j$ is the total number of fermions.\n\nSince the spin system is on a ring, there can be only an even number of domain walls in the system. As noticed above, the operator for the occupation number by a domain wall $\\widehat{n}_j$ coincides with the number of fermions at site $j$, $f^\\dag_j f_j$. Therefore we have to consider the restriction $\\widehat{\\mathbb{M}}_+(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) $ of $\\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ to the sector with an even number of fermions.\nAccording to \\eqref{widehatM} the expression of this rectriction is invariant by translation along the ring if the fermionic operators are chosen to satisfy the antiperiodic boundary conditions\n\\begin{equation}\n\\label{AntiperiodicC}\nf_{2N+1}=-f_1 \\quad\\textrm{and}\\quad f^\\dag_{2N+1}=-f^\\dag_1.\n\\end{equation}\nThen the restriction reads\n\\begin{equation}\n\\label{Mplus}\n\\frac{2}{\\nu_1+\\nu_2} \\widehat{\\mathbb{M}}_+(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) \n=-(1-\\gamma) N \\mathbb{1} -\\gamma\\sum_{j=1}^{2N}f^\\dag_jf_j\n+\\sum_{j=1}^{2N} \\overline{\\nu}_j\\left[f^\\dag_j f_{j+1} - f_jf^\\dag_{j+1} +c_j f^\\dag_jf^\\dag_{j+1} -b_j f_jf_{j+1} \\right].\n\\end{equation}\n\n\\clearpage\n\\subsection{Antiperiodic Fourier transform}\n\nThe next step to the diagonalization is to rewrite the fermionic operators as antiperiodic Fourier transforms which satisfy the antiperiodic boundary conditions \\eqref{AntiperiodicC}\nThe wave numbers are of the form $q=(2k+1)\\pi\/(2N)$ and we work with a complete family of representatives in the set \n\\begin{equation}\n\\label{defBod2N}\n{\\cal B}(2N)=\\{q=(2k+1) \\frac{\\pi}{2N}, k=-N,-N+1, \\cdots, -1,0,1, \\cdots N-1 \\},\n\\end{equation}\nnamely\n\\begin{equation}\n{\\cal B}(2N)=\\{-\\pi +\\frac{\\pi}{2N}, -\\pi +\\frac{3\\pi}{2N}, \\cdots -\\frac{\\pi}{2N}, \\frac{\\pi}{2N}, \\cdots \\pi-\\frac{\\pi}{2N} \\}.\n\\end{equation}\nThe operator $f_j$ can be written as the antiperiodic Fourier transform\n\\begin{equation}\n\\label{defAntiFT}\nf_j=\\frac{1}{\\sqrt{2N}} \\sum_{q\\in {\\cal B}(2N)} \\text{\\mdseries e}^{\\imath qj} \\eta_q\n\\end{equation}\nin terms of the wave fermions\n\\begin{equation}\n\\label{defAntiFTInv}\n\\eta_q =\\frac{1}{\\sqrt{2N}} \\sum_{j=1}^{2N} \\text{\\mdseries e}^{-\\imath qj} f_j.\n\\end{equation}\nGoing from \\eqref{defAntiFT} to \\eqref{defAntiFTInv} relies on the identity\n\\begin{equation}\n\\label{SumGeom}\n\\sum_{j=1}^{2N} \\text{\\mdseries e}^{\\imath(q-q')j}=2N \\,\\mathbb{1}_{q-q'\\equiv 0 (2\\pi)},\n\\end{equation}\nwhere $\\mathbb{1}_{q-q'\\equiv 0 (2\\pi)}=1$ if $q-q'$ is equal to $0$ modulo $2\\pi$ and $\\mathbb{1}_{q-q'\\equiv 0 (2\\pi)}=0$ otherwise.\nAll $q'$s in ${\\cal B}(2N)$ satisfy $\\text{\\mdseries e}^{\\imath q 2N}=-1$, and subsequently $f_j$ does obey the antiperiodic boundary conditions \\eqref{AntiperiodicC}.\nThe adjoint operator $f^\\dag_j$ reads\n\\begin{equation}\nf^\\dag_j=\\frac{1}{\\sqrt{2N}} \\sum_{q\\in {\\cal B}(2N)} \\text{\\mdseries e}^{-\\imath qj} \\eta^\\dag_q.\n\\end{equation}\n\nThese representations are inserted in the expression \\eqref{Mplus}. In the term $\\sum_{j=1}^{2N}f^\\dag_jf_j$ there occurs a summation over all sites of the ring and one uses the identity \\eqref{SumGeom}.\n In the other summations one has to distinguish the two sublattices ; for instance one has to consider the sum $\\sum_{n=1}^N f^\\dag_{2n} f^\\dag_{2n+1}$ and then one uses the identity\n\\begin{equation}\n\\sum_{n=1}^{N} \\text{\\mdseries e}^{\\imath(q+q')2n}=N \\,\\mathbb{1}_{2(q+q')\\equiv 0 (2\\pi)}=N \\,\\mathbb{1}_{q+q'\\equiv 0(\\pi)}.\n\\end{equation}\nAccording to definition \\eqref{defBod2N}, the set ${\\cal B}(2N)$ does not contain the value $0$ and the solution of $q+q'=0$ corresponds to two distinct values $q$ and $-q$. \n\n\nIn order to simplify the following discussion, we assume from now on that $N$ \\textit{is even}. Then ${\\cal B}(2N)$ does not contain $\\pi\/2$ and all values $q$ and $\\pi- q$, are also distinct. \nAfter a symmetrization over the values $q$ and $\\pi-q$ the matrix $\\widehat{\\mathbb{M}}_+(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ appears as a sum of contributions each of which involves only the operators associated with the wave numbers $q$, $\\pi-q$, $-q$ and $-(q-\\pi)$. Let us introduce \nthe first quadrant in the set ${\\cal B}$ defined as\n\\begin{equation}\n\\label{defQBod}\n{\\cal QB}(2N)=\\{q=(2k+1) \\frac{\\pi}{2N}, k=0,1, \\cdots (N\/2) -1 \\} =\\{ \\frac{\\pi}{2N}, \\frac{3\\pi}{2N}\\cdots \\frac{\\pi}{2}-\\frac{\\pi}{2N}\\}.\n\\end{equation}\nThen the expression \\eqref{Mplus} for\n$\\widehat{\\mathbb{M}}_+(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ can be rewritten as\n\\begin{equation}\n\\label{widehatMBis}\n\\frac{2}{\\nu_1+\\nu_2} \\widehat{\\mathbb{M}}_+(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) \n=-N \\mathbb{1} +\\sum_{q\\in {\\cal QB}} \\left[V^\\dag_q \\right]^{\\scriptscriptstyle T} \\mathbb{A}_q(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})\\, V_q,\n\\end{equation}\nwhere $V_q$ is the column vector\n\\begin{equation}\n\\label{defVq}\nV_q=\n\\begin{pmatrix}\n\\eta_q \\\\\n\\eta_{q-\\pi} \\\\\n\\eta^\\dag_{-q} \\\\\n\\eta^\\dag_{\\pi-q}\n\\end{pmatrix},\n\\end{equation}\n$\\left[V^\\dag_q \\right]^{\\scriptscriptstyle T}$ denotes the transposed row vector corresponding to the column vector $V^\\dag_q$ built with the adjoints of the components of $V_q$, and\n\\begin{equation}\n\\mathbb{A}_q (\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) =\n\\begin{pmatrix}\n -\\gamma + \\cos q & \\imath a_q \\sin q & \\imath c'_q \\sin q & c_q \\cos q \\\\\n - \\imath a_q \\sin q & -\\gamma - \\cos q & -c_q \\cos q & -\\imath c'_q \\sin q \\\\\n- \\imath b'_q \\sin q & - b_q \\cos q & \\gamma - \\cos q & -\\imath a_q \\sin q \\\\\nb_q \\cos q & \\imath b'_q \\sin q & \\imath a_q \\sin q & \\gamma + \\cos q\n\\end{pmatrix},\n\\end{equation}\nwith \n\\begin{eqnarray}\na_q &=& \\overline{\\nu}_\\text{\\mdseries o}-\\overline{\\nu}_\\text{\\mdseries e}\n\\\\ \\nonumber\nb_q&=& \\overline{\\nu}_\\text{\\mdseries o} b_\\text{\\mdseries o} -\\overline{\\nu}_\\text{\\mdseries e} b_\\text{\\mdseries e}\n\\\\ \\nonumber\nb'_q&=& \\overline{\\nu}_\\text{\\mdseries o} b_\\text{\\mdseries o} +\\overline{\\nu}_\\text{\\mdseries e} b_\\text{\\mdseries e} \n\\\\ \\nonumber\nc_q&=& \\overline{\\nu}_\\text{\\mdseries o} c_\\text{\\mdseries o} -\\overline{\\nu}_\\text{\\mdseries e} c_\\text{\\mdseries e}\n\\\\ \\nonumber\nc'_q&=& \\overline{\\nu}_\\text{\\mdseries o} c_\\text{\\mdseries o} +\\overline{\\nu}_\\text{\\mdseries e} c_\\text{\\mdseries e},\n\\end{eqnarray}\nwhere the $b_\\text{\\mdseries a}$'s and the $c_\\text{\\mdseries a}$'s are defined in \\eqref{defbj} and \\eqref{defcj}. \n\n\\subsection{Diagonalization of $\\mathbb{A}_q(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e}) $}\n\nThe characteristic polynomial of $\\mathbb{A}_q$, $\\text{det}\\left(\\mathbb{A}_q-\\alpha \\mathbb{1}_q\\right)$, proves to be a second order polynomial in $\\alpha^2$, with a constant which is a squared quantity,\n\\begin{equation}\n\\label{CharacteristicPol}\n\\text{det}\\left(\\mathbb{A}_q-\\alpha \\mathbb{1}_q\\right)= \\alpha^4 -2 D \\alpha^2 +F^2.\n\\end{equation}\nMoreover both coefficients $D$ and $F^2$ depend on the parameters $\\lambda_\\text{\\mdseries o}$ and $\\lambda_\\text{\\mdseries e}$ only through the difference\n\\begin{equation}\n\\label{deflambdab}\n\\overline{\\lambda}=\\left(\\lambda_\\text{\\mdseries e}-\\lambda_\\text{\\mdseries o}\\right)\\Delta E.\n\\end{equation}\nThey read\n\\begin{equation}\nD=1 + (\\overline{\\nu}_\\text{\\mdseries o}-\\overline{\\nu}_\\text{\\mdseries e})^2 + \\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\left[4 \\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}\\cos^2q + (1-2\\cos^2q) \\theta(\\overline{\\lambda}) \\right]\n\\end{equation}\nand \n\\begin{equation}\nF=\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\left[ 4(1- \\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e} \\cos^2 q) + \\theta(\\overline{\\lambda})\\right],\n\\end{equation}\nwhere the function $\\theta(\\overline{\\lambda})$ vanishes when $\\overline{\\lambda}$ is set to zero :\n\\begin{equation}\n\\label{deftheta}\n\\theta(\\overline{\\lambda})=2\\left[ (1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}) (\\cosh \\overline{\\lambda} -1) + (\\gamma_\\text{\\mdseries o}-\\gamma_\\text{\\mdseries e})\\sinh\\overline{\\lambda} \\right].\n\\end{equation}\nThe coefficient $F$ can be rewritten as \n\\begin{equation}\nF=\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\left[ 2 + 2 \\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}(1-2 \\cos^2 q) + (1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}) \\cosh \\overline{\\lambda} + (\\gamma_\\text{\\mdseries o}-\\gamma_\\text{\\mdseries e})\\sinh\\overline{\\lambda}\\right],\n\\end{equation}\nand the property $(1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}) > \\vert \\gamma_\\text{\\mdseries o}-\\gamma_\\text{\\mdseries e} \\vert$ for $\\gamma_\\text{\\mdseries o}<1$ and $\\gamma_\\text{\\mdseries e}<1$ ensures that $F>0$.\nThe squared roots of the characteristic polynomial are\n\\begin{equation}\n\\label{valuealpha2}\n\\alpha^2=D \\pm \\sqrt{D^2-F^2} = \\left(\\sqrt{\\frac{D+F}{2}} \\pm \\sqrt{\\frac{D-F}{2}} \\right)^2.\n\\end{equation}\nwhere $\\sqrt{\\cdots}$ denotes a possibly complex square root. Let us introduce the notations\n$R_\\pm(q,\\overline{\\lambda})= \\tfrac{1}{2}(D\\pm F)$, namely\n\\begin{equation}\n\\label{defRplus}\nR_+(q,\\overline{\\lambda})= 1+\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\theta(\\overline{\\lambda}) \\sin^2q\n\\end{equation}\n\\begin{equation}\n\\label{defRminus}\nR_-(q,\\lambda)=(\\overline{\\nu}_\\text{\\mdseries o}-\\overline{\\nu}_\\text{\\mdseries e})^2+\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\left[4 \\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e} -\\theta(\\overline{\\lambda})\\right]\\cos^2 q.\n\\end{equation}\nWe notice that, since $F$ is positive, $R_+> R_-$.\n\nNote that $1+\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\theta(\\overline{\\lambda})>0$ because of the definition \\eqref{deftheta} of $\\theta(\\overline{\\lambda})$ and the identities\n $1-2\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} (1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}) >0$ and $(1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e})> \\vert \\gamma_\\text{\\mdseries o}-\\gamma_\\text{\\mdseries e} \\vert$\n for $\\gamma_\\text{\\mdseries o}<1$ and $\\gamma_\\text{\\mdseries e}<1$. According to \\eqref{defRplus}, $R_+$ can be rewritten as \n$\nR_+=1-\\sin^2 q + \n\\left(1+\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\theta(\\overline{\\lambda})\\right)\\sin^2q\n$ and we conclude that \n$R_+>0$. \n\nOn the other hand, by virtue of definitions \\eqref{defgammaa} and \\eqref{deftheta},\n \\begin{equation}\n \\label{relthetacosh}\n4 \\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e} -\\theta(\\overline{\\lambda})=2\\frac{\\cosh\\left((\\beta_\\text{\\mdseries e}+\\beta_\\text{\\mdseries o}) \\Delta E\/2\\right)-\\cosh\\left(\\overline{\\lambda}-(\\beta_\\text{\\mdseries e}-\\beta_\\text{\\mdseries o}) \\Delta E\/2\\right)}{\\cosh\\left(\\beta_\\text{\\mdseries e} \\Delta E\/2\\right)\\cosh\\left(\\beta_\\text{\\mdseries o} \\Delta E\/2\\right)},\n\\end{equation}\nso that $4 \\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e} -\\theta(\\overline{\\lambda})>0$ only if $-\\beta_\\text{\\mdseries o} \\Delta E<\n\\overline{\\lambda}<\\beta_\\text{\\mdseries e}\\Delta E$, and we infer from \\eqref{defRminus} that $R_-$ can take both signs. \n\nEventually the four eigenvalues of $\\mathbb{A}_q(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ are\n\\begin{equation}\n\\label{defalphan}\n\\alpha_1= \\sqrt{R_+} + \\sqrt{R_-} \\qquad \\alpha_2= \\sqrt{R_+} - \\sqrt{R_-}\\qquad \\alpha_3=-\\alpha_2 \\qquad \\alpha_4=-\\alpha_1,\n\\end{equation}\nIn these expressions $\\sqrt{R_+}$ denotes the usual positive square root of the positive number $R_+$, whereas $\\sqrt{R_-}$ is either real positive or purely imaginary depending on the sign of $R_-$, i.e. on the values of $\\overline{\\lambda}$ and $q$.\n As noticed previously $R_+> R_-$, and in the case where $\\sqrt{R_-}$ is real, all $\\alpha_k$'s, with $k=1,\\ldots,4$, are real and $\\alpha_1>\\alpha_2>0> \\alpha_3 > \\alpha_4$.\n\n\n\n\n\n\\subsection{Largest eigenvalue of the modified matrix $\\widehat{\\mathbb{M}}_+(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$}\n\nFor the sake of conciseness we omit all dependences upon $\\lambda_\\text{\\mdseries o}$ and $\\lambda_\\text{\\mdseries e}$ in the present section.\nWe denote by $\\mathbb{D}_q$ the diagonal matrix built with the eigenvalues $\\alpha_1(q),\\ldots,\\alpha_4(q)$ of the matrix $\\mathbb{A}_q(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$. The matrix $\\mathbb{A}_q$ reads\n\\begin{equation}\n\\label{relAD}\n\\mathbb{A}_q= \\mathbb{P}_q \\mathbb{D}_q \\mathbb{P}_q^{-1},\n\\end{equation}\nwhere the $k^\\text{th}$ column of $\\mathbb{P}_q$ is made with the components of a (column) right eigenvector of \n$\\mathbb{A}_q$ associated with the eigenvalue $\\alpha_k(q)$, and $\\mathbb{P}_q^{-1}$ is the inverse matrix of $\\mathbb{P}_q$. Let $\\xi_k(q)$ denote the $k^\\text{th}$ component of the column vector $\\mathbb{P}_q^{-1} V_q$, while $\\xi^\\star_k(q)$ denotes the \n$k^\\text{th}$ component of the row vector $\\left[V^\\dag_q \\right]^{\\scriptscriptstyle T} \\mathbb{P}_q$, where $V_q$ and $\\left[V^\\dag_q \\right]^{\\scriptscriptstyle T} $ are defined in \\eqref{defVq}. With these definitions, the relation \\eqref{relAD} implies that\n\\begin{equation}\n\\label{Sumxi}\n\\left[V^\\dag_q \\right]^{\\scriptscriptstyle T} \\mathbb{A}_q V_q=\\sum_{k=1}^4 \\alpha_k(q) \\, \\xi^\\star_k(q) \\,\\xi_k(q).\n\\end{equation}\nThe operators $\\xi_k$ and $\\xi^\\star_k$ obey the anticommutation rules $\\{\\xi_k, \\xi_{k'}\\}=0$, $\\{\\xi^\\star_k, \\xi^\\star_{k'}\\}=0$ and $\\{\\xi_k, \\xi^\\star_{k'}\\}=\\delta_{k,k'}$. However, since $\\mathbb{A}_q$ is not hermitian (for the usual scalar product), \n$\\mathbb{P}_q$ is not unitary and the operator $\\xi^\\star_k$ is not the adjoint of $\\xi_k$.\n\n\nNevertheless the anticommutation rules are enough to ensure that the spectrum of the operator $\\xi^\\star_k \\xi_k$ is the set $\\{0,1\\}$. Then, according to the expressions \\eqref{defalphan} of the $\\alpha_k$'s, the value of the right-hand side of \\eqref{Sumxi} with the largest real part is equal to the sum of two eigenvalues and proves to be real positive,\n\\begin{equation}\n\\alpha_1 + \\alpha_2=2 \\sqrt{R_+(q)}>0.\n\\end{equation}\nEventually, according to \\eqref{widehatMBis}, the largest eigenvalue of $ \\widehat{\\mathbb{M}}_+(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ is\n\\begin{equation}\n\\label{LargestEigenvalue}\n\\frac{\\nu_\\text{\\mdseries o}+\\nu_\\text{\\mdseries e}}{2}\n\\left[-N + 2\\sum_{q\\in {\\cal QB}} \\sqrt{R_+(q)}\\right],\n\\end{equation}\nwhere $R_+(q)$ is given by \\eqref{defRplus}.\n\nWe notice that if $\\lambda_\\text{\\mdseries o}=\\lambda_\\text{\\mdseries e}=0$, the modified Markov matrix $\\widehat{\\mathbb{M}}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ becomes the usual Markov matrix for the evolution of $P(\\mathbf{n};t)$; since $\\theta(0)=0$, we retrieve that the largest eigenvalue of the Markov matrix is $0$. Moreover the eigenvalue closest to 0 is obtained by setting $ \\xi^\\star_3(q)\\xi_3(q)$ equal to 1 for $q=0$, and we retrieve the value \\eqref{trel} for the inverse relaxation time of the magnetizations on the two sublattices.\n\n\\clearpage\n\\section{Cumulants of heat amounts per unit of time in the long time limit}\n\\label{SecCumulants}\n\n\\subsection{Scaled generating function for joint cumulants}\n\nAccording to the remark at the end of section \\ref{CumulantsMaximumEigenvalue}, the scaled generating function for the joint cumulants of $\\Heat_\\text{\\mdseries o}$ and $\\Heat_\\text{\\mdseries e}$ coincides with the largest eigenvalue of $ \\widehat{\\mathbb{M}}_+(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$. By virtue of \\eqref{LargestEigenvalue} it reads\n\\begin{equation}\n\\label{ExpgL}\ng_{2N}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})= \\frac{\\nu_\\text{\\mdseries o}+\\nu_\\text{\\mdseries e}}{2}\\left[-N + 2 \\sum_{q\\in {\\cal QB}} \\sqrt{1+\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\theta(\\overline{\\lambda}) \\sin^2q}\\right],\n\\end{equation}\nwhere ${\\cal QB}(2N)$ is defined in \\eqref{defQBod} and $\\theta(\\overline{\\lambda})$ is given in \\eqref{deftheta}.\nThe joint cumulants per unit of time in the long time limit are determined from the relation\n\\begin{equation}\n\\label{JointCumulant}\n\\lim_{t\\to\\infty} \\frac{1}{t}\\Esp{\\Heat_\\text{\\mdseries e}^p \\Heat_\\text{\\mdseries o}^{p'}}_c =\n\\left.\\frac{\\partial^{p+p'} g_{2N}(\\lambda_\\text{\\mdseries e},\\lambda_\\text{\\mdseries o};t)}{\\partial \\lambda_\\text{\\mdseries e}^p \\partial \\lambda_\\text{\\mdseries o}^{p'}}\\right\\vert_{\\lambda_\\text{\\mdseries e}=\\lambda_\\text{\\mdseries o}=0},\n\\end{equation}\nwhere the index $c$ refers to the truncation of the mean value $\\Esp{\\Heat_\\text{\\mdseries e}^p \\Heat_\\text{\\mdseries o}^{p'}}$ involved in the definition of the cumulant.\nThe fact that $g_{2N}$ depends only on the difference $\\lambda_\\text{\\mdseries e}-\\lambda_\\text{\\mdseries o}$ entails the properties\n\\begin{equation}\n\\lim_{t\\to\\infty} \\frac{1}{t}\\Esp{\\Heat_\\text{\\mdseries e}^p \\Heat_\\text{\\mdseries o}^{p'}}_c =(-1)^{p'}\\lim_{t\\to\\infty} \\frac{1}{t}\\Esp{\\Heat_\\text{\\mdseries e}^{p+p'}}_c \n\\end{equation}\nand, in particular,\n\\begin{equation}\n\\lim_{t\\to\\infty} \\frac{1}{t}\\Esp{\\Heat_\\text{\\mdseries o}^p}_c =(-1)^{p}\\lim_{t\\to\\infty} \\frac{1}{t}\\Esp{\\Heat_\\text{\\mdseries e}^p}_c.\n\\end{equation}\nThese properties are linked to the fact that the interface energy can take only a finite number of values whereas the cumulants have no upper bounds in the infinite time limit.\n\nFor the sake of completeness we point out that, according to \\eqref{relthetacosh}, $\\theta(\\overline{\\lambda})$ depends on $\\overline{\\lambda}$ through the function \n$\\cosh\\left(\\overline{\\lambda}-(\\beta_\\text{\\mdseries e}-\\beta_\\text{\\mdseries o})\\Delta E\/2\\right)$; therefore the scaled generating function satisfies the symmetry $g_{2N}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})=g_{2N}(\\beta_\\text{\\mdseries o}-\\lambda_\\text{\\mdseries o}, \\beta_\\text{\\mdseries e}-\\lambda_\\text{\\mdseries e})$, which is in fact a consequence of the local detailed balance \\eqref{LDB}. Since $g_{2N}(\\lambda_\\text{\\mdseries o},\\lambda_\\text{\\mdseries e})$ depends only on the difference $\\lambda_\\text{\\mdseries e}-\\lambda_\\text{\\mdseries o}$, this entails a symmetry for the scaled generating function for the cumulants of $\\Heat_\\text{\\mdseries e}$, $g^\\text{e}_{2N}(\\lambda_\\text{\\mdseries e})=g_{2N}(0,\\lambda_\\text{\\mdseries e})$, namely the symmetry $g^\\text{e}_{2N}(\\lambda_\\text{\\mdseries e})=g^\\text{e}_{2N}(\\beta_\\text{\\mdseries e}-\\beta_\\text{\\mdseries o}-\\lambda_\\text{\\mdseries e})$. Then the corresponding large deviation function for the time-integrated current ${\\cal J}_\\text{\\mdseries e}=\\Heat_\\text{\\mdseries e}\/t$, which can be obtained as the Legendre-Fenchel transform of $g^\\text{e}_{2N}(\\lambda_\\text{\\mdseries e})$, obeys the fluctuation relation $f({\\cal J}_\\text{\\mdseries e})-f(-{\\cal J}_\\text{\\mdseries e})=(\\beta_\\text{\\mdseries o}-\\beta_\\text{\\mdseries e}){\\cal J}_\\text{\\mdseries e}$.\n\n\n\\subsection{Cumulants for heat amount $\\Heat_\\text{\\mdseries e}$}\n\nWe give the explicit expressions for the first four cumulants of $\\Heat_\\text{\\mdseries e}$ per unit of time. They are determined as $\\partial^n g_{2N}(0,\\lambda_\\text{\\mdseries e}) \/ \\partial\\lambda_\\text{\\mdseries e}^n\\vert_{\\lambda_\\text{\\mdseries e}=0}=\\Delta E^n \\times \\partial^n g_{2N}(0,\\lambda_\\text{\\mdseries e}) \/ \\partial\\overline{\\lambda}^n\\vert_{\\overline{\\lambda}=0}$, where $\\overline{\\lambda}=\\left(\\lambda_\\text{\\mdseries e}-\\lambda_\\text{\\mdseries o}\\right)\\Delta E$ and $\\Delta E=4K$ is the energy gap in the chain energy. The function $\\theta(\\overline{\\lambda})$ defined in \\eqref{deftheta} can be rewritten as\n\\begin{equation}\n\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} \\theta(\\overline{\\lambda})= 2A [\\cosh \\overline{\\lambda} -1] + 2B \\sinh \\overline{\\lambda}\n\\end{equation}\nwhere the parameters $A$ and $B$ are those introduced in Ref.\\cite{CornuBauer2013} for a model with only two spins, namely\n\\begin{equation}\nA=\\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} (1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e})\n\\end{equation}\n\\begin{equation}\nB= \\overline{\\nu}_\\text{\\mdseries o}\\overline{\\nu}_\\text{\\mdseries e} (\\gamma_\\text{\\mdseries o}-\\gamma_\\text{\\mdseries e}).\n\\end{equation}\nThen, according to \\eqref{ExpgL}, the first four cumulants of $\\Heat_\\text{\\mdseries e}$ read\n\\begin{eqnarray}\n\\label{ExpCumulants}\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}}}{(\\nu_\\text{\\mdseries e}+\\nu_\\text{\\mdseries o})t}&=& \\frac{N}{2} B S_2 \\times \\Delta E\n\\\\\n\\nonumber\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}^2}_c}{(\\nu_\\text{\\mdseries e}+\\nu_\\text{\\mdseries o})t}&=& \\frac{N}{2} \\left[A S_2-B^2S_4\\right] \\times \\Delta E^2\n\\\\\n\\nonumber\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}^3}_c}{(\\nu_\\text{\\mdseries e}+\\nu_\\text{\\mdseries o})t} &=& \\frac{N}{2} B \\left[ S_2- 3 A S_4+3B^2 S_6\\right] \\times \\Delta E^3\n\\\\\n\\nonumber\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}^4}_c}{(\\nu_\\text{\\mdseries e}+\\nu_\\text{\\mdseries o})t} &=& \\frac{N}{2} \\left[A S_2- (3 A^2 + 4B^2) S_4 +18 A B^2 S_6 -15 B^4 S_8 \\right] \\times \\Delta E^4,\n\\end{eqnarray}\nwhere\n\\begin{equation}\nS_{2n}(N)=\\frac{2}{N}\\sum_{k=0}^{(N\/2)-1} \\sin^{2n} \\left(\\frac{(2k+1) \\pi}{2N} \\right).\n\\end{equation}\nThe structure of the cumulants in terms of the coefficients $A$ and $B$ is similar to the structure found for the two-spin model of Ref.\\cite{CornuBauer2013} as well as for the ring of Ref.\\cite{CornuHilhorst2017}; indeed, in the three cases the scaled cumulant generating function depends on $\\lambda_\\text{\\mdseries e}$ only through the same function $\\theta(\\lambda_\\text{\\mdseries e} \\Delta E)$.\n\nThe coefficients $S_{2n}(N)$ (with $N$ even) can be calculated explicitly. In the special case $N=2$, a direct calculation leads to $S_{2n}(2)=(1\/2)^n$. For any $N\\geq 2$,\n by extending the summation up to $N$, rewriting $\\sin u=[\\text{\\mdseries e}^{\\imath u}-\\text{\\mdseries e}^{-\\imath u}]\/(2\\imath)$, using the binomial formula and an identity similar to \\eqref{SumGeom}, we get that\n\\begin{equation}\n\\label{Wallis1}\n\\textrm{if $n4$ the first four cumulants are given by \\eqref{ExpCumulants} where the $S_{2n}(N)$ are to be replaced by the corresponding $W_{2n}$ given in \\eqref{valueWallis}.\n\n\nEventually the dependences upon the thermodynamic parameters, $\\gamma_\\text{\\mdseries o}$ and $\\gamma_\\text{\\mdseries e}$, and the kinetic parameters, $\\nu_\\text{\\mdseries o}$ and $\\nu_\\text{\\mdseries e}$, arise only through the coefficients $A$ and $B$. This is in contrast with what happens for another Ising chain model, where both thermostats act on every spin \\cite{CornuHilhorst2017}: for this model the dependence upon the combination $\\gamma$ of the thermodynamic and kinetic parameters defined in \\eqref{defgamma} also arises in the coefficients $\\Sigma_n(N,\\gamma)$ which replace the coefficients $S_n(N)$ in the expressions \\eqref{ExpCumulants} for the cumulants.\n\n\\clearpage\n\\subsection{Various physical regimes}\n\nAccording to the last remark of the previous section, the discussion of the various physical regimes is the same as that performed in the case of the two-spin model of Ref.\\cite{CornuBauer2013}.\n\nAt equilibrium $\\gamma_\\text{\\mdseries o}=\\gamma_\\text{\\mdseries e}$ and, according to \\eqref{deftheta}, $\\theta(\\overline{\\lambda})=2(1-\\gamma_\\text{\\mdseries e}^2) (\\cosh \\overline{\\lambda} -1)$. Therefore only cumulants of even order do not vanish. According to \\eqref{ExpCumulants} the first two cumulants of even order read\n\\begin{eqnarray}\n\\label{ExpCumulantsEq}\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}^2}_c}{(\\nu_\\text{\\mdseries e}+\\nu_\\text{\\mdseries o})t}&=& \\frac{1}{4} \\nu_\\text{\\mdseries o}\\nu_\\text{\\mdseries e} \\left( 1-\\gamma_\\text{\\mdseries e}^2\\right) \\times N \\Delta E^2\n\\\\\n\\nonumber\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}^4}_c}{(\\nu_\\text{\\mdseries e}+\\nu_\\text{\\mdseries o})t} &=& \\frac{1}{4} \\nu_\\text{\\mdseries o}\\nu_\\text{\\mdseries e} \\left( 1-\\gamma_\\text{\\mdseries e}^2\\right) \\left[1-\\frac{9}{4}\\nu_\\text{\\mdseries o}\\nu_\\text{\\mdseries e} \\left( 1-\\gamma_\\text{\\mdseries e}^2\\right) \\right]\\times N\\Delta E^4.\n\\end{eqnarray}\nThe probability distribution of $\\Heat_\\text{\\mdseries e}$ is not a Gaussian, since all cumulants of even order have non-vanishing values.\n\n\\medskip\n\nWhen a thermostat has a kinetic parameter far larger than the other one, the scaled generating function becomes proportional to $\\theta(\\overline{\\lambda})$, \n\\begin{equation}\ng_{2N}(\\overline{\\lambda})=\\frac{1}{8} N \\nu_\\text{\\mdseries s}\\theta(\\overline{\\lambda}),\n\\end{equation}\nwhere $\\nu_\\text{\\mdseries s}$ is the kinetic parameter of the slower thermostat. As a consequence $g_{2N}(\\overline{\\lambda})$\ncoincides with the scaled generating function of a continuous-time random walk, because $\\theta(\\overline{\\lambda})$ can be rewritten as\n\\begin{equation}\n\\theta(\\overline{\\lambda})= 2 \\left[ p_+ \\text{\\mdseries e}^{\\overline{\\lambda}} +p_- \\text{\\mdseries e}^{-\\overline{\\lambda}} -(p_++p_-)\\right]\n\\end{equation}\nwith the probabilities $p_+=(1+\\gamma_\\text{\\mdseries o})(1-\\gamma_\\text{\\mdseries e})\/2$ and $p_-=(1-\\gamma_\\text{\\mdseries o})(1+\\gamma_\\text{\\mdseries e})\/2$. As a consequence all cumulants of even (odd) order are equal to the same value when they are measured in unit of $\\Delta E$: for all $p\\geq 1$\n\\begin{eqnarray}\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}^{2p-1}}_c}{ t \\, \\Delta E^{2p-1}} &=& \\frac{1}{4} \\nu_\\text{\\mdseries s} (\\gamma_\\text{\\mdseries o}-\\gamma_\\text{\\mdseries e}) \\, N\n\\\\\n\\nonumber\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}^{2p}}_c}{t \\, \\Delta E^{2p}} &=& \\frac{1}{4} \\nu_\\text{\\mdseries s} (1-\\gamma_\\text{\\mdseries o}\\gamma_\\text{\\mdseries e}) \\, N.\n\\end{eqnarray}\nIn the same kinetic regime, if one thermostat is at zero temperature, for instance $\\gamma_\\text{\\mdseries o}=1$, then\nthe scaled generating function coincides with that of a continuous-time Poisson process, because\n\\begin{equation}\n\\theta(\\overline{\\lambda})= 2 (1-\\gamma_\\text{\\mdseries e}) \\left[ \\text{\\mdseries e}^{\\overline{\\lambda}}-1\\right].\n\\end{equation}\nAs a consequence all cumulants of $\\Heat_\\text{\\mdseries e}$ in unit of $\\Delta E$ are equal to the same value: for all $p\\geq 1$\n\\begin{equation}\n\\lim_{t\\to\\infty} \\frac{\\Esp{\\Heat_\\text{\\mdseries e}^{p}}_c}{t \\, \\Delta E^p} = \\frac{1}{4} \\nu_\\text{\\mdseries s} (1-\\gamma_\\text{\\mdseries e}) \\, N.\n\\end{equation}\n\n\\section{Conclusion}\n\n\nIn the present paper we have investigated the heat currents in an Ising spin ring where alternating spins are coupled to two macroscopic bodies at different temperatures and with different kinetic parameters. The stationary mean values of the global two-spin correlations at any distance have been calculated. The dependence upon the kinetic parameters arises only through the linear combination $\\gamma$ defined in \\eqref{defgamma}\\footnote{When the kinetic parameters are set equal, our result are compatible with those of Ref.\\cite{SchmittmannSchmuser2002}.}. The finite-size corrections in the global two-spin correlations disappear in the mean instantaneous global heat current coming out of one thermostat.\n\nThe scaled generating function of the joint cumulants per unit of time for the heat amounts exchanged with the two thermostats over a long time have been calculated exactly. At leading order in the ring size they prove to be proportional to the ring size, as it is the case for the model where two thermostats act on the same site \\cite{CornuHilhorst2017}.\n Moreover, if the order of the cumulant is lower than the number of spins connected to one thermostat, the finite-size corrections again disappear exactly, and the cumulant is strictly proportional to the ring size.\n\n We notice that the proportionality to the ring size at leading order in the size has already been observed for the cumulants of two other kinds of cumulative quantities when the system is homogeneous (only one temperature and one kinetic parameter) and seen as a Simple Exclusion Process with pair creation and annihilation \\cite{PopkovSchutz2011}: the two cumulative quantities are the difference between the numbers of domain wall jumps in the clockwise and anticlockwise directions respectively, and the number of pair annihilations. This is in contrast with the case of the purely diffusive Simple Exclusion Process on a ring, where the cumulants for the difference between the numbers of jumps in the two directions and the cumulants for the total number of jumps are proportional to powers of the ring size which increase with the order of the cumulants \\cite{AppertDerridaETAL2008}.\n\n\nAn interesting open problem is the calculation of the heat cumulants in another solvable model for thermal contact : two joined semi-infinite Ising chains \ncoupled to thermostats at two different temperatures \\cite{LavrentovichZia2010,Lavrentovich2012}. Then the mean global current which flows from one thermostat to the other through the junction between the two half-chains is obtained by summing the mean currents received by all spins in a semi-infinite Ising chain. The intrinsic inhomogeneity of these currents\nwould have to be dealt with by specific methods.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nUnstable particles (UP's) are described usually by dressed propagator or with the help of the $S$-matrix with complex pole. The problems of these descriptions have been under considerable discussion for many decades\n[\\refcite{1}]--[\\refcite{7}] (and references therein). There are\nalso other approaches such as time-asymmetric QFT of UP's\n[\\refcite{6}], effective theory [\\refcite{8,9}], modified\nperturbative theory approach [\\refcite{10}], and phenomenological\nQF model of UP's with smeared mass [\\refcite{11,12}]. In this\nwork we consider some remarkable properties of the model [\\refcite{11,12}] which\nare caused by mass smearing and lead to the factorization effects\nin the description of the processes with UP in an intermediate\nstate.\n\nThe model under consideration is based on the time-energy\nuncertainty relation (UR). Despite their formal uniformity,\nvarious UR's have different physical nature. This point has been\ndiscussed for many years beginning with Heisenberg's formulating\nthe uncertainty principle (for instance, see\n[\\refcite{13}]--[\\refcite{16}] and references therein). The first\nmodel of UP based on the time-energy UR was suggested in\n[\\refcite{17}]. Time-dependent wave function of UP in its rest\nframe was written in terms of the Fourier transform which may be\ninterpreted as a distribution of mass values, with a spread,\n$\\delta m$, related to the mean lifetime $\\delta \\tau =1\/\\Gamma$\nby uncertainty relation [\\refcite{14,17}]:\n\\begin{equation}\\label{1.1}\n\\delta m\\cdot\\delta\\tau\\sim 1,\\,\\,\\,\\mbox{or}\\,\\,\\,\\delta m\\sim\n\\Gamma\\,\\,\\,(c=\\hbar =1).\n\\end{equation}\nThus, from the time-energy UR for the unstable quantum system, we\nare led to the concept of mass smearing for UP, which is described\nby UR (\\ref{1.1}). Implicitly (indirectly) the time-energy UR, or\ninstability, is usually taken into account by using the complex\npole in $S$-matrix or dressed propagator which describes UP in an\nintermediate state. Explicit account of the relation (\\ref{1.1})\nis taken by describing UP in a final or initial states with the\nhelp of the mass-smearing effect. From Eq. (\\ref{1.1}) it follows\nthat this effect is noticeable if UP has a large width. Mass\nsmearing was considered in the various fields of particle\nphysics---in the decay processes of UP with large width\n[\\refcite{12}], in the boson-pair production [\\refcite{18,19}],\nand in the phenomenon of neutrino oscillations [\\refcite{16,20}].\nIn these papers, the efficiency of the mass-smearing conception\nwas demonstrated in a wide class of processes.\n\nThe effects of exact factorization in the two-particle scattering\nand three-particle decay were shown for the cases of scalar,\nvector, and spinor UP in Refs.~[\\refcite{21,22}]. Factorized\nformulae for cross section and decay width were derived exactly\n(without any approximations) for the tree-level processes in the\nphysical gauge. However, the results can be generalized taking\naccount of principal part of radiative corrections\n[\\refcite{12,18,19,21}] (see Section 3). The factorization method\nthat is based on the exact factorization in the simplest processes\nwas suggested in [\\refcite{22a}]. In this work we generalize the\nresults of [\\refcite{22a}] to the case of UP with spin $J=3\/2$ and\napply them to some complicated processes. In the second section we\ndescribe the main elements of the model that lead to the effects\nof factorization. These effects in the case of the simplest\nprocesses are considered in Section 3 for UP with\n$J=0,\\,1\/2,\\,1,\\,3\/2$. The factorization method based on the results\nof the third section is applied to some more\ncomplicated processes in Section 4. In this section, we also\nconsider the accuracy of the calculations. Some conclusions are\nmade concerning the applicability and advantages of the method in\nthe fifth section.\n\n\\section{Model of UP with a smeared mass}\n\n\nIn this section we present the elements of the model\n[\\refcite{11,12}] that are used directly in the factorization\nmethod. The field function of the UP is a continuous superposition\nof the standard ones defined at a fixed mass with a weight\nfunction of the mass parameter $\\omega(\\mu)$ which describes mass\nsmearing. As a result of this, an amplitude of the process with\nUP in a final or initial state has the form:\n \\begin{equation}\\label{2.1}\n A(k,\\mu)=\\omega(\\mu)A^{st}(k,\\mu),\n \\end{equation}\nwhere $A^{st}(k,\\mu)$ is an amplitude defined in a standard way at\nfixed mass parameter $\\mu$, and $\\omega(\\mu)$ is a model weight\nfunction.\n\nThe model Green's function has a convolution form with respect to\nthe mass parameter $\\mu$ (the Lehmann representation). In the case\nof scalar UP it is as follows:\n \\begin{equation}\\label{2.2}\n D(x)=\\int D(x,\\mu)\\,\\rho(\\mu)\\,d\\mu,\\,\\,\\,\\rho(\\mu)=|\\omega(\\mu)|^2,\n \\end{equation}\nwhere $D(x,\\mu)$ is defined in a standard way for a fixed\n$\\mu = m^2$ and $\\rho(\\mu)$ is a probability density of the mass parameter\n$\\mu$.\n\nThe model propagators of scalar, vector, and spinor unstable\nfields in momentum representation are given by the following\nexpressions [\\refcite{12}]:\n\\begin{align}\\label{2.3}\n D(q)=&i\\int\\frac{\\rho(\\mu)\\,d\\mu}{q^2-\\mu+i\\epsilon},\\,\\,\\,D_{mn}(q)=-i\\int\\frac{g_{mn}-\n q_{m}q_{n}\/\\mu}{q^2-\\mu+i\\epsilon}\\rho(\\mu)\\,d\\mu,\\notag\\\\\n &\\hat{G}=i\\int\\frac{\\hat{q}+\\sqrt{\\mu}}{q^2-\\mu+i\\epsilon}\\rho(\\mu)\\,d\\mu,\n \\qquad q=\\sqrt{(q_iq^i)}.\n\\end{align}\nThe model propagators are completely defined if the function $\\rho(\\mu)$ is\ndetermined.\n\nHere, we generalize the results of the works [\\refcite{12,21,22}]\nto include the case of unstable fields with spin $J=3\/2$. The propagator of\nthis field is defined in [\\refcite{23,24}] and smearing its mass, $M^2\\to\\mu$, gives:\n\\begin{align}\\label{2.4}\n &\\hat{G}_{mn}(q)=\\int\\rho(\\mu)\\,d\\mu \\{\n -\\frac{\\hat{q}+\\sqrt{\\mu}}{q^2-\\mu+i\\epsilon} ( g_{mn} -\\frac{1}{3} \\gamma_{m}\n \\gamma_{n} - \\frac{\\gamma_{m} q_{n} -\\gamma_{n} q_{m}}{3\\sqrt{\\mu}} - \\frac{2}{3}\n \\frac{q_{m} q_{n}}{\\mu})\\}.\n\\end{align}\n\nDetermination of the weight function $\\omega(\\mu)$ or\ncorresponding probability density $\\rho(\\mu)=|\\omega(\\mu)|^2$ can\nbe done with the help of the various methods [\\refcite{12}]. Here\nwe consider the definition of $\\rho(\\mu)$ which leads to the\neffect of exact factorization. We match the model propagator of\nscalar UP to the standard dressed one:\n\\begin{equation}\\label{2.5}\n \\int\\frac{\\rho(\\mu)d\\mu}{k^2-\\mu+i\\epsilon}\\longleftrightarrow\n \\frac{1}{k^2-M^2_0-\\Pi(k^2)}\\,,\n\\end{equation}\nwhere $\\Pi(k^2)$ is a conventional polarization function. It was\nshown in [\\refcite{11,12}] that the correspondence (\\ref{2.5})\nleads to the definition:\n\\begin{equation}\\label{2.6}\n \\rho(\\mu)=\\frac{1}{\\pi}\\,\\frac{Im\\Pi(\\mu)}{[\\mu-M^2(\\mu)]^2+[Im\\Pi(\\mu)]^2}\\,,\n\\end{equation}\nwhere $M^2(\\mu)=M^2_0+Re\\Pi(\\mu)$. Substitution of the expression\n(\\ref{2.6}) into (\\ref{2.3}) and integration over $\\mu$ lead to\nthe results:\n\\begin{equation}\\label{2.7}\n D_{mn}(q)=i\\frac{-g_{mn}+q_m\n q_n\/q^2}{q^2-M^2(q^2)-iIm\\Pi(q^2)}\n\\end{equation}\nand\n\\begin{equation}\\label{2.8}\n \\hat{G}(q)= i\\frac{\\hat{q}+q}{q^2-M^2(q^2)-iq\\Sigma(q^2)}\\,.\n\\end{equation}\nIn analogy with these definitions we get the expression for the\npropagator of vector-spinor unstable field:\n\\begin{equation}\\label{2.9}\n \\hat{G}_{mn}(q)=\n -\\frac{\\hat{q}+q}{q^2-M^2(q^2)-iq\\Sigma(q^2)} \\{ g_{mn} -\\frac{1}{3} \\gamma_{m}\n \\gamma_{n} - \\frac{\\gamma_{m} q_{n} -\\gamma_{n} q_{m}}{3q} - \\frac{2}{3}\n \\frac{q_{m} q_{n}}{q^2}\\}.\n\\end{equation}\nNote that in Eqs. (\\ref{2.8}) and (\\ref{2.9}) we have substituted\n$q\\Sigma(q^2)$ for $\\Pi(q^2)$. The expressions\n(\\ref{2.5})--(\\ref{2.9}) define an effective theory of UP's which\nfollows from the model and the definition (\\ref{2.6}), that is\nfrom the correspondence (\\ref{2.5}). In this theory the numerators\nof the expressions (\\ref{2.7})--(\\ref{2.9}) differ from the\nstandard ones. The correspondence between standard and model\nexpressions for the cases of vector (in unitary gauge) and spinor\nUP is given by the interchange $m \\leftrightarrow q$ in the\nnumerators of the standard and model propagators (here\n$q=\\sqrt{q_i q^i})$. As a result the structures of the model\npropagators lead to the effect of exact factorization, while the\nstandard ones lead to approximate factorization (see the next\nsection).\n\n\\section{Effects of factorization in the processes\\\\ with UP in an intermediate state}\n\nExact factorization is stipulated by the following properties of the model:\\\\\na) smearing the mass shell of UP in accordance with the time-energy UR;\\\\\nb) specific structure of the numerators of the propagators, Eqs.~(\\ref{2.7})--(\\ref{2.9}).\\\\\nThe first factor allows us to describe UP in an intermediate state\nwith the momentum $q$ as a particle in a final or initial states\nwith the variable mass $m^2=q^2$. In this case, UP is described by\nthe following polarization matrices which differ from the standard\non-shell ones by the change $m\\to q$ (see also [\\refcite{12}]):\n\\begin{align}\\label{3.1}\n&\\sum_{a=1}^{3} e^a_m(\\vec{q})\\dot{e}^a_n(\\vec{q})=-g_{mn}+\\frac{q_m q_n}{q^2}\\,\\,\\,\\,\\,\\mbox{(vector UP)};\\notag\\\\\n&\\sum_{a=1}^{2}u^{a,\\mp}_i(\\vec{q})\\bar{u}^{a,\\pm}_k(\\vec{q})=\\frac{1}{2q^0}(\\hat{q}+q)_{ik}\n\\,\\,\\,\\,\\,\\mbox{(spinor UP)};\\notag\\\\\n&\\hat{\\Pi}_{mn}(q)=-\\frac{1}{4} ( \\hat{q} + q ) \\left[g_{mn}\n-\\frac{1}{3} \\gamma_{m} \\gamma_{n}\n - \\frac{\\gamma_m q_n -\\gamma_n q_m}{3q} - \\frac{2}{3} \\frac{q_m q_n}{q^2}\n \\right],\\notag\\\\\n &\\mbox{(vector-spinor UP)}.\n\\end{align}\nThe second factor is the coincidence of the expressions for the\npropagator numerators (\\ref{2.7})--(\\ref{2.9}) and for the\npolarization matrices (\\ref{3.1}). It allows us to represent the\namplitude of the process with UP in an intermediate state (see\nFig.1) in a partially factorized form:\n\\begin{equation}\\label{3.2}\n\\mathit{M}(p,p',q)=K\\sum_{a}\\frac{\\mathit{M}^{(a)}_1(p,q)\\cdot\n\\mathit{M}^{(a)}_2(p',q)}{P(q^2,M^2)},\n\\end{equation}\nwhere $\\mathit{M}^{(a)}$ is a spiral amplitude. The representation\n(\\ref{3.2}) is a precondition of factorization, while full exact\nfactorization occurs in the transition probability. The effect of\nfactorization is illustrated in Fig.1 where UP in an intermediate\nstate is signed by crossed line.\n\n\\begin{figure}[!ht]\n \\centerline{\\epsfig{file=factor1.eps,height=2cm,width=5cm}}\n \\caption{Factorization in the reducible diagram.}\n \\label{fig:Born1}\n\\end{figure}\n\n\nNow, we demonstrate the factorization effect in the case of the\nsimplest basic elements of the tree processes, where UP is in the\n$s-$ channel intermediate state. The vertices are described by the\nsimplest standard Lagrangians for scalar, vector, and spinor\nparticles (see Refs. [\\refcite{21,22}]). Here we add the\nLagrangian that describes the interaction of vector-spinor\nparticles with lower-spin ones [\\refcite{25}]--[\\refcite{27}]:\n\\begin{align}\\label{3.3}\n \\mathscr{L}_{int}&=\n \\frac{f}{m_\\pi} \\bar{\\Psi}^\\mu \\bar{\\Theta}_{\\nu\\mu} (z,\\lambda) N \\pi^{,\\nu}_a +\n \\mbox{h.c.}\\notag\\\\\n &+\\frac{g}{2 M_N} \\bar{\\Psi}^\\mu \\bar{\\Theta}_{\\nu\\mu} (x,\\lambda)\n \\gamma^{\\xi} \\gamma_5 N\\rho_{\\xi}{}^{\\nu} +\n \\mbox{h.c.},\n\\end{align}\nwhere\n\\begin{equation}\\label{3.4}\n \\Theta_{\\mu\\nu} (\\lambda,\\lambda') =\n g_{\\mu\\nu} + \\frac{\\lambda'-\\lambda}{2\\left( 2 \\lambda - 1 \\right)} \\gamma_\\mu\n \\gamma_\\nu\\,,\n\\end{equation}\n$x$, $z$ are off-shell parameters, and $\\lambda$ is the parameter of\nthe Lagrangian of the Rarita-Schwinger free field.\n\nLet us consider the simplest processes which are used further as the\nbasic elements of the method. Note that calculations are\nmade at the \"tree level\" in the framework of the effective\ntheory, which follows from the model with mass smearing and contains self-energy\ncorrections in the function $\\rho(\\mu)$. This effective theory is\nnot a gauge one, but the definitions of the model propagators\n(\\ref{2.3}) are given in analogy with the standard physical\ngauge (unitary gauge for massive fields).\n\nThe first element is two-particle scattering with UP of any type\nin the intermediate state.\n\\begin{figure}[!ht]\n \\centerline{\\epsfig{file=factor2.eps,height=2cm,width=5cm}}\n \\caption{\\footnotesize Factorization in $2\\to 2$ scattering diagram}\n \\label{fig:Born2}\\end{figure}\n\nBy straightforward calculation at the tree level it was shown that\nthe cross-section for all permissible combinations of particles\n$(a,b,R,c,d)$ can be represented in the universal factorized form\n[\\refcite{21}]:\n\\begin{equation}\\label{3.5}\n \\sigma(ab\\rightarrow R\\rightarrow cd)=\\frac{16\\pi(2J_R+1)}\n {(2J_a+1)(2J_b+1)\\bar{\\lambda}^2(m_a,m_b;\\sqrt{s})}\n \\frac{\\Gamma^{ab}_R(s)\\Gamma^{cd}_R(s)}{|P_R(s)|^2}.\n\\end{equation}\nHere, $s=(p_a+p_b)^2$, $P_R(s)$ is the denominator of the\npropagator of an unstable particle $R(s)$ which is defined by\nEqs.~ (\\ref{2.7}) and (\\ref{2.8}), $\\lambda(m_a,m_b;\\sqrt{s})$ is\nnormalized K\\\"{a}ll\\'{e}n function and\n$\\Gamma^{ab}_R(s)=\\Gamma(R(s)\\to ab)$ is a partial width of the\nparticle $R(s)$ with variable mass $m=\\sqrt{s}$. Note that exact\nfactorization in the process under consideration always takes\nplace for scalar $R$ in both the standard and model treatment. In\nthe case of vector and spinor $R$ the factorization is exact in\nthe framework of the model only. In the standard treatment the\nexpression (\\ref{3.5}) is valid in the narrow-width approximation\n(see, for instance, Eq.~(37.51) and corresponding comment in\n[\\refcite{27a}]). If the diagram depicted in Fig.2 is included into a more\ncomplicated one as a sub-diagram, where the particle $c$ and\/or $d$\nare unstable, then we have to generalize partial width, for\ninstance, $\\Gamma^{cd(q)}_R (s,q)=\\Gamma(R(s)\\to cd(q))$, where\n$d(q)$ is unstable particle $d$ with mass $m^2=q^2$ (see the next\nsection). Note that Eq.(\\ref{3.5}) differs from the corresponding\nequation (6) in [\\refcite{21}] and equation (42) in [\\refcite{22}]\nwhich contain misprints. Correct expressions can be got in\n[\\refcite{21,22}] by modification $k_a k_b\/k_R\\to k_R\/k_a k_b$,\nwhere $k_p=2J_p+1$ and $J_p$ is spin of the particle $p =\na,\\,b,\\,R$.\n\nIt was shown in Ref.[\\refcite{21}] that Eq.(\\ref{3.5}) is valid\nfor the cases of scalar ($J=0$), vector ($J=1$), and spinor\n($J=1\/2$) unstable particles. In this work we check by direct\ncalculation that Eq.(\\ref{3.5}) is also correct in the case of UP\nwith $J=3\/2$, when Eq.~(\\ref{2.9}) is used (for instance,\n$\\Delta$-resonance production). We should note that the\nexpressions which involve vector-spinor UP are valid for the\nparticles on the mass shell in the framework of the standard\ntreatment. However, in the framework of the model, UP is always on\nits smeared mass shell and these expressions are valid in general\ncase. Moreover, the part of expressions which contains off-shell\nparameters as well as $\\lambda$-parameter disappears in widths and\ncross-sections. Hence the condition of factorization---the\ncoincidence of the polarization matrix and the numerators of the\npropagators---is fulfilled in the case under consideration (see\nEqs. (\\ref{2.9}) and (\\ref{3.1})).\n\nThe second basic element is a three-particle decay with UP in the\nintermediate state $\\Phi\\to\\phi_1R\\to\\phi_1\\phi_2\\phi_3$ (Fig.\\ref{fig:Born3}), where\n$R$ is UP of any kind.\n\\begin{figure}[!ht]\n \\centerline{\\epsfig{file=factor3.eps,height=3cm,width=5cm}}\n \\caption{\\footnotesize Factorization in $1\\to 3$ decay diagram}\n \\label{fig:Born3}\\end{figure}\n\nBy straightforward calculations it was shown that the\nthree-particle partial width at the tree level can be represented\nin the universal factorized form [\\refcite{22}]:\n\\begin{equation}\\label{3.6}\n \\Gamma(\\Phi\\rightarrow\\phi_1 \\phi_2 \\phi_3)=\\int_{q^2_1}^{q^2_2}\\Gamma(\\Phi\n \\rightarrow\\phi_1 R(q))\\,\\frac{q\\,\\Gamma(R(q)\\rightarrow\\phi_2\\phi_3)}\n {\\pi \\vert P_{R}(q)\\vert ^2}\\,dq^2\\,,\n\\end{equation}\nwhere $R$ is a scalar, vector or spinor UP, $q_1=m_2+m_3$ and\n$q_2=m_{\\Phi}-m_1$. By direct calculations we also check the\nvalidity of the expression (\\ref{3.6}) for the case of the UP with\n$J=3\/2$ (see the remark to Eq.(\\ref{3.5})). It is seen clearly that\nthe formula (\\ref{3.6}) can include any factorizable corrections.\n\nBy summing over decay channels of $R$, from\nEq.(\\ref{3.6}) we get the well-known convolution formula for the\ndecays with UP in a final state [\\refcite{22}]:\n\\begin{equation}\\label{3.7}\n \\Gamma(\\Phi\\rightarrow\\phi_1 R) = \\int_{q^2_1}^{q^2_2} \\Gamma(\\Phi\\rightarrow\n \\phi_1 R(q))\\,\\rho_{R}(q)\\,dq^2\\,.\n\\end{equation}\nIn Eq.(\\ref{3.7}) smearing the mass of unstable state $R$ is\ndescribed by the probability density $\\rho_R(q)$:\n\\begin{equation}\\label{3.8}\n \\rho_{R}(q)=\\frac{q\\,\\Gamma^{tot}_R(q)}{\\pi\\,\\vert P_{R}(q)\\vert^2}.\n\\end{equation}\nThe expression (\\ref{3.8}) is connected with Eq.(\\ref{2.6}) by the\nrelation $Im\\Pi(q)=q\\, \\Gamma^{tot}_R(q)$.\n\nThe factorized expressions (\\ref{3.6}) and (\\ref{3.7}) are\napplied successfully for the description of the decay $B\\to \\rho D$\n[\\refcite{12}], decay properties of $\\phi(1020)$-meson\n[\\refcite{12}] and $t$-quark [\\refcite{31,32,33}], lightest\nchargino and next-to-lightest neutralino [\\refcite{33a}].\nMoreover, the formula (\\ref{3.6}) describes\nthe decays $\\mu\\to e\\bar{\\nu}_e \\nu_{\\mu}$, $\\tau\\to\ne\\bar{\\nu}_e\\nu_{\\tau}$, and $\\tau^-\\to \\nu_{\\tau}\\pi^-\\pi^0$ with great accuracy (see\nthe next section). It should be noted that, in analogy with two-particle\nscattering, the factorization in the expression for the width\n(\\ref{3.6}) is also exact within the framework of the model and\napproximate in the standard treatment (convolution method). Besides,\nwe note that the expressions (\\ref{3.5}) and (\\ref{3.6})\nsignificantly simplify calculations in comparison with the standard\nones.\n\n\\section{Factorization method in the model of UP's\\\\ with smeared mass}\n\nThe method is based on exact factorization of the simplest\nprocesses with UP in an intermediate state that were considered in\nSection 3. The factorization method has applicability to such Feynman diagrams that can be disconnected into two components by cutting some line corresponding to timelike momentum transfer. For instance, it is applicable to the\ncomplicated scattering and decay-chain processes which can be\nreduced to a chain of the basic elements (\\ref{3.5}) and\n(\\ref{3.6}). Next, we consider some examples of such processes.\n\n\\begin{equation}\\label{4.1}\n1)\\, a+b\\to R_1\\to c+R_2\\to c+d+f \\text{ (Fig.\\ref{fig:Born4})}.\n\\end{equation}\n\\begin{figure}[!ht]\n \\centerline{\\epsfig{file=factor4.eps,height=3cm,width=5cm}}\n \\caption{\\footnotesize Factorization in $2\\to 3$ scattering-decay diagram}\n \\label{fig:Born4}\\end{figure}\n\nThe cross-section of this process is a combination of the\nexpressions (\\ref{3.5}) and (\\ref{3.6}):\n\\begin{align}\\label{4.2}\n &\\sigma(ab\\to R_1\\to cdf)=\\notag\\\\\n &\\frac{16k_{R_1}}{k_a k_b\\bar{\\lambda}^2(m_a,m_b;\\sqrt{s})}\\frac{\\Gamma^{ab}_{R_1}(s)}{|P_{R_1}(s)|^2}\n \\int_{q^2_1}^{q^2_2}\\Gamma(R_1(s)\\to\n c R_2(q))\\,\\frac{q\\,\\Gamma^{df}_{R_2}(q)}{|P_{R_2}(q)|^2}\\,dq^2,\n\\end{align}\nwhere $k_p=2J_p+1$. It should be noted that the factorization\neffectively reduces the number of independent kinematic\nvariables which have to be integrated. In the standard approach\nfor the process $2\\to 3$ the number of the variables, which\nuniquely specify a point in the phase space, in the general case\nis $N=3n-4=5$, from which four variables have to be integrated\n[\\refcite{28}]. Some of this variables can be integrated out if a\nspecific symmetry of the process occur. In the framework of the\napproach suggested the number of integrated variables is always\n$N_M=1$. The same effect of variable reduction takes place for the\ncase of the basic processes of scattering and decay which were\nconsidered in the previous section.\n\nThe expression (\\ref{4.2}) can be used for fast evaluation\nof cross sections of some scattering processes both in cosmology\nand collider physics. For instance, it is valid for the\ndescription of the annihilation process with the lightest\nsupersymmetric particle in a final state [\\refcite{33a}].\n\n\\begin{equation}\\label{4.3}\n2)\\, \\Phi\\to a+R_1\\to a+b+R_2\\to a+b+c+d\\text{ (Fig.\\ref{fig:Born5})}.\n\\end{equation}\n\\begin{figure}[!ht]\n \\centerline{\\epsfig{file=factor5.eps,height=3cm,width=5cm}}\n \\caption{\\footnotesize Factorization in $1\\to 4$ decay diagram}\n \\label{fig:Born5}\\end{figure}\n\nThe width of this decay-chain process is given by doubling the\nformula (\\ref{3.6}):\n\\begin{align}\\label{4.4}\n\\Gamma(\\Phi\\to abcd)&=\\frac{1}{\\pi^2}\\int_\n{q^2_1}^{q^2_2}\\frac{q\\,\\Gamma(\\Phi\n \\rightarrow a R_1(q))}{\\vert P_{R_1}(q)\\vert ^2}\\times\\notag\\\\&\\times\\int_{g^2_1}^{g^2_2}\\Gamma(R_1(q)\\to bR_2(g))\n \\,\\frac{g\\,\\Gamma(R_2(g)\\rightarrow cd)}\n {\\vert P_{R_2}(g)\\vert ^2}\\,dg^2\\,dq^2.\n\\end{align}\nNote that in the general case of $n$-particle decay the number of\nkinematic variables, which uniquely specify a point in the phase\nspace, is $N=3n-7=5$ [\\refcite{28}], while the\nmethod gives $N_M=2$ (see comment to the previous case). Thus, we\nhave a significant simplification of calculations. Some\nexamples of processes which can be described by the compact\nformula (\\ref{4.4}) are considered in [\\refcite{33}], where the\nrelation between convolution method and decay-chain method is\nanalyzed.\n\n\\begin{equation}\\label{4.5}\n3)\\, a+b\\to c+R\\to c+d+e\\text{ (Fig.\\ref{fig:Born6})}.\n\\end{equation}\n\n\\begin{figure}[!ht]\n \\centerline{\\epsfig{file=factor6.eps,height=3cm,width=5cm}}\n \\caption{Factorization in $a+b\\to c+R\\to c+d+e$ process.}\n \\label{fig:Born6}\n\\end{figure}\n\nThe cross-section of this $t$-channel process is described by\nconvolution of the cross-section $\\sigma(ab\\to cR)$ and the width\n$\\Gamma(R\\to de)$:\n\\begin{equation}\\label{4.6}\n\\sigma(ab\\to cde)=\\frac{1}{\\pi}\\int_{q_1^2}^{q_2^2}\\sigma(ab\\to\ncR(q))\\frac{q\\Gamma(R(q)\\to de)}{\\vert P_{R}(g)\\vert ^2}\\,dq^2.\n\\end{equation}\nThis formula can be applied to the description of the processes\n$e^+e^-\\to \\gamma Z\\to \\gamma f\\bar{f}$ and $eN\\to e\\Delta \\to\ne\\pi N$. The diagram in Fig.6 illustrates a class of processes at\nthe tree level with fermion-antifermion pair in the one-pole\napproximation, i.e. generated from the decay of $R$ only. However,\nthe formula (\\ref{4.6}) can be easily generalized taking account\nof factorizable radiative corrections. For instance, such a\ngeneralization of the expression (\\ref{4.6}) was made in\n[\\refcite{28a}] for the description of the process $e^+e^-\\to\n\\gamma Z\\to \\gamma \\sum_f \\nu_f\\bar\\nu_f$, $f=e,\\,\\mu,\\,\\tau$, with $\\nu\\bar\\nu$-pairs being produced\nby $Z$-decay only (``single-pole'' resonant production). Note that additional non-resonant ``ladder'' diagrams also contribute to the process $e^+e^-\\to\n\\gamma \\nu\\bar\\nu$. The resonant events can be still separated in certain kinematic regions [\\refcite{28aa}]. In our calculations [\\refcite{28a}] (with the kinematic cut corresponding to the event selection of Ref.[\\refcite{28aa}]) we have taken into account ISR and principal part\nof radiative corrections. These corrections do not change the\nstructure of the expression (\\ref{4.6}) and satisfy the condition\nof factorization. The results are in good agreement with the\nexperimental data and SM predictions at\n$\\sqrt{s}=185 - 210\\,\\mbox{GeV}$ [\\refcite{28a}]. We should note,\nhowever, that our calculations are valid in the energy domain, where\nthe selection of the resonant events is possible.\n\n\\begin{equation}\\label{4.7a}\n4)\\, e^+e^-\\to ZZ\\to \\sum_{i,k} \\bar{f}_i f_i \\bar{f}_k f_k\\text{\n(Fig.\\ref{fig:Born7})}.\n\\end{equation}\n\n\\begin{figure}[!ht]\n \\centerline{\\epsfig{file=factor7.eps,height=2cm,width=5cm}}\n \\caption{$Z$-pair production process.}\n \\label{fig:Born7}\n\\end{figure}\nNow, we consider the process of $Z$-pair production (or\nfour-fermion production in the double-pole approximation). Direct\napplication of the model to the process $e^+e^-\\to ZZ$ or using\nthe factorization method for the full process $e^+e^-\\to\nZZ\\to \\sum_{i,k} \\bar{f}_i f_i \\bar{f}_k f_k$ (double-pole\napproach) gives the following expression for cross-section at the\ntree level [\\refcite{18}]:\n\\begin{equation}\\label{4.8a}\n\\sigma^{tr}(e^+e^-\\to ZZ)=\\int\\int\\sigma^{tr}(e^+e^-\\to\nZ_1(m_1)Z_2(m_2))\\,\\rho_Z(m_1)\\,\\rho_Z(m_2)\\,dm_1\\,dm_2,\n\\end{equation}\nwhere $\\sigma^{tr}(e^+e^-\\to Z_1(m_1)Z_2(m_2))$ is defined in\na standard way for the case of fixed boson masses $m_1$ and $m_2$,\nand probability density of mass $\\rho(m)$ is defined by the\nexpression:\n\\begin{equation}\\label{4.7}\n\\rho_Z(m)=\\frac{1}{\\pi}\\,\\frac{m\\,\\Gamma^{tot}_Z(m)}{(m^2-M^2_Z)^2+(m\\,\\Gamma^{tot}_Z(m))^2}.\n\\end{equation}\nSimilar expressions can be written for the processes $e^+e^-\\to\nW^+W^-$ [\\refcite{19}] and $e^+e^-\\to ZH$ [\\refcite{28a}]. To\ndescribe exclusive processes, such as $e^+e^-\\to ZZ\\to\nf_i\\bar{f}_i f_k\\bar{f}_k$, one has to substitute partial\n$q$-dependent width $\\Gamma_Z^i(q)=\\Gamma(Z(q)\\to f_i\\bar{f}_i)$\ninto the expression (\\ref{4.7}) instead of the total width\n$\\Gamma^{tot}_Z(m)$. Note that the expression similar to\n(\\ref{4.6}) can be written in the standard approach as a result of\nintegration over the phase space variables which describe\n$4f$-states in the semi-analytical approximation (SAA)\n[\\refcite{28b}]. Within the framework of the model considered,\nformula (\\ref{4.6}) is derived exactly without any approximations.\nNote also that as a rule the deviation of the standard exact\nresults from the model ones is negligible (see Eq. (\\ref{4.12})).\n\nThe processes $e^+e^-\\to ZZ, W^+W^-, \\gamma Z, ZH$ were considered\nin detail [\\refcite{17,18,28a}] taking account\nof the relevant radiative corrections. It was shown that the\nresults of the model calculations are in good agreement with the\nexperimental LEP II data and coincide with standard Monte-Carlo\nresults with great accuracy. At the same time the factorization\nmethod significantly simplifies the calculation procedures in\ncomparison with the standard ones that should consider about ten thousands\ndiagrams (see, for example, [\\refcite{28c}]\nand comments in [\\refcite{18}]).\n\nUsing the factorization method one can describe complicated decay-chain and scattering\nprocesses in a simple way. The same results can occur within the\nframe of standard treatment as the approximations. Such\napproximations are known as narrow-width approximation (NWA)\n[\\refcite{29,30}], convolution method (CM) [\\refcite{31}]--[\\refcite{33}], decay-chain method (DCM) [\\refcite{33}] and\nsemi-analytical approach (SAA) [\\refcite{28b}]. All these\napproximations get a strict analytical formulation within the\nframework of the factorization method. For instance, NWA includes five assumptions which\nwas considered in detail in [\\refcite{29}]. The factorization method\ncontains just one assumption---non-factorizable corrections are small\n(the fifth assumption of NWA). The method suggested can be applied\nto very complicated decay-chain and scattering processes by combining the expressions considered above. In this case\nwe have not a strict and general standard analog of such\napproximation.\n\nNow, we consider some ways of evaluation of the method error which\nwe define as the deviation of the model results from the strict\nstandard ones. For a scalar UP the error always equals zero in\naccordance with the definition (\\ref{2.5}). For a vector UP the\nerror is caused by the following difference:\n\\begin{equation}\\label{4.8}\n\\delta\\eta_{\\mu\\nu}=\\eta_{\\mu\\nu}(q^2)-\\eta_{\\mu\\nu}(m^2)=q_{\\mu}q_{\\nu}\\frac{m^2-q^2}{m^2q^2},\n\\end{equation}\nwhere $\\eta_{\\mu\\nu}(m^2)$ and $\\eta_{\\mu\\nu}(q^2)$ are standard\nand model numerators of vector propagators in the physical gauge. In\nthe case of meson-pair production $e^+e^-\\to\n\\rho^0,\\omega,...\\to\\pi^+\\pi^-, K^+K^-,\\rho^+\\rho^-,...$ the\ndeviation equals zero too, due to vanishing contribution of the\ntransverse parts of the amplitudes in both cases:\n\\begin{equation}\\label{4.9}\n\\mathit{M}^{trans}(q)\\sim\n\\bar{e}^-(p_1)\\hat{q}e^-(p_2)=\\bar{e}^-(p_1)(\\hat{p}_1+\\hat{p}_2)e^-(p_2)=0.\n\\end{equation}\nIn the case of the high-energy collisions $e^+e^-\\to Z\\to\nf\\bar{f}$ (we neglect $\\gamma-Z$ interference) the transverse part\nof the amplitude is:\n\\begin{equation}\\label{4.10}\n\\mathit{M}^{trans}(q)\\sim\n\\bar{e}^-(p_1)\\hat{q}(c_e-\\gamma_5)e^-(p_2)\\cdot\n\\bar{f}^+(k_1)(c_f-\\gamma_5)f^+(k_2)\n\\end{equation}\nand we get at $q^2\\approx M^2_Z$:\n\\begin{equation}\\label{4.11}\n\\delta \\mathit{M}\\sim \\frac{m_em_f}{M^2_Z}\\frac{M_Z-q}{M_Z}.\n\\end{equation}\nThus, an error of the factorization method at the vicinity of resonance is always small,\nmoreover, it is suppressed by small factor $m_e m_f\/M^2_Z$. The\nsimilar estimations can be easily done for the case of a spinor\nUP.\n\nThe relative deviation of the model cross section of the\nboson-pair production with consequent decay of the bosons to\nfermion pairs is [\\refcite{18}]:\n\\begin{equation}\\label{4.12}\n\\epsilon_f \\sim\n4\\frac{m_f}{M}[1-M\\int_{m^2_f}^{s}\\frac{\\rho(q^2)}{q}\\,dq^2],\n\\end{equation}\nwhere $M$ is a boson mass. For the case $f=\\tau$ a deviation is\nmaximal, $\\epsilon_{\\tau}\\sim 10^{-3}$. It should be noted that\nthe deviations that are caused by the approach at the tree level\nare significantly smaller then the errors caused by the\nuncertainty in taking account of radiative corrections\n[\\refcite{19}]. Thus, the the error of the method at the tree\nlevel in the case of vector UP is, as a rule, negligible.\n\nBy straightforward calculations we evaluate the relative\ndeviations of the model partial width from standard one, that is,\n$\\epsilon =(\\Gamma^M-\\Gamma^{st})\/\\Gamma_M$ for the case of $\\mu$\nand $\\tau$ decays. We get:\n\\begin{equation}\\label{4.13}\n\\epsilon(\\mu\\to e\\nu\\bar{\\nu})\\approx 5\\cdot\n10^{-4};\\,\\,\\,\\epsilon(\\tau\\to e\\nu\\bar{\\nu})\\approx 3\\cdot\n10^{-6};\\,\\,\\,\\epsilon(\\tau\\to \\mu\\nu\\bar{\\nu})\\approx 3\\cdot\n10^{-2}.\n\\end{equation}\nThe deviation is suppressed by the factor $k=m^2_{l1}\/m^2_{l2}$,\nwhich is small in the first and second case and large in the last\ncase. In the case of the decay $\\tau^-\\to \\nu_{\\tau}\\pi^-\\pi^0$,\nsuppression factor is very small,\n$k=(m^2_{\\pi_0}-m^2_{\\pi_-})^2\/m^4_{\\tau} \\sim 10^{-7}$.\n\nIn the case of a spinor UP in an intermediate state a deviation is\nof the order of $(M_f-q)\/M_f$. It can be large when $q$ is far\nfrom the resonance region. However, in analogy with vector UP,\nthis deviation can be suppressed by small factor too, and we have\nto control this effect in every case under consideration. The same\neffect can occur in the case of the vector-spinor UP.\n\n\\section{Conclusion}\n\nThe model of UP's leads to effective theory of UP's with a\nspecific structure of vector, spinor and vector-spinor\npropagators. Such a structure gives rise to the effects of exact\nfactorization in a broad class of the processes with UP's in\nintermediate states. These effects allow us to develop the\nfactorization method for the description of the complicated\nprocesses with participation of an arbitrary type of UP's. The\nmethod suggested is simple and convenient tool for deriving the\nformulae for cross sections and decay rates in the case of\ncomplicated scattering and decay-chain processes. The\nfactorization method can be used as some analytical analog of NWA,\nwhich enables us to evaluate the error of the approach at the tree\nlevel in a simple way. We have shown that these errors, as a rule,\nare significantly smaller then the ones caused by the uncertainty\nin taking account of radiative corrections. It should be noted that the applicability of the method is limited to the energy scales where the non-resonant or non-factorizable contributions can be neglected.\n\nThe factorization method based on the model of UP with a smeared\nmass can be treated in two various ways. On the one hand, it\nfollows from the specific structure of propagators and can be\ninterpreted as some heuristic (irrespective of the model) way to\nevaluate decay rates and cross sections easily with the help of\nthe concise and convenient expressions. On the other hand, the\nmodel, from which the factorization method follows, is based on the fundamental\nproperties of UP---time-energy UR (i.e., smearing the mass). Thus,\nthe method can be also used as some physical basis for development\nof precision tools of rapid and easy calculations.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n\n\\emph{Neural architecture search} (NAS) is a rapidly growing area of \\emph{machine learning} (ML) \ndedicated to automatically designing high performing \\emph{deep learning} models.\nRecent breakthroughs, such as \\textit{differentiable search}, e.g., DARTS \\cite{DARTS},\nhave enabled search at limited computing cost and time.\nHowever, state-of-the-art methodologies still suffer from limited interpretability,\nand current evaluation protocols do not always shed light on the contribution of individual components \n(i.e., search space, training pipeline) while reporting performances \\cite{Yang2020NasEvalHard,Lindauer2020BestPractices}.\n\n\nRecently, a \\emph{fitness landscape analysis}-based (FLA) methodology was introduced: the \\emph{Fitness Landscape Footprint} (a.k.a., \\emph{footprint}) \\cite{traore2021fitness}.\nThe \\emph{footprint} attempts to describe \nwhy a search strategy may be successful, struggle or fail on a target application.\nIt also enables comparing search problems of variable configuration \n(i.e., different search space, fitness function, data, etc.).\n\nOur study takes advantage of the \\emph{footprint} to identify the most favorable sensor setting \nfor NAS.\nParticularly, we consider optimizing convolutional neural network (CNN) image classifiers on the search space defined by NASBench-101 \\cite{ying2019nasbench101} for the real-world image classification problem So2Sat LCZ42~\\cite{So2SatDataset}.\nOur results show that disregard the sensor, the longer the training time, the better the performance (fitness) and the flatter the landscape (less ruggedness and deviation in fitness). \nMoreover, Sentinel-2 and fusion (Sentinel-1 and 2) tend to have more favorable search trajectories (smoother, higher persistence).\nTo the best of our knowledge, our study provides the first quantification \nand comparison of search behavior across sensors (including sensor fusion).\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width=0.5\\textwidth]{Figures\/footprint\/footprint_various_36.png}\n \\caption{\\emph{Fitness landscape footprint} of So2Sat LCZ42 for Sentinel-1 (blue), Sentinel-2 (yellow), and both sensors together (green), on CNN image classifiers NAS problem.} \n \\label{fig:footprnts}\n\\end{figure}\n\nThis article is structured as follows: \nNext section summarizes the related work and Section~\\ref{sec:fitness-landscape} introduces the \\emph{footprint}. \nSection~\\ref{sec:footprint-for-fusion} proposes the methodology to study NAS problems. \nSections \\ref{sec:exp-setup} and~\\ref{sec:results} presents the experimental settings and the results.\nSection~\\ref{sec:conclusion} outlines the conclusions and proposes future work.\n\n\n\n\n\n\n\n\\section{Related Work}\\label{sec:related-work}\n\n\\emph{Computer vision} (CV) and \\emph{Earth observation} (EO) are closely tied~\\cite{ball2017comprehensive,zhu2017deep}.\nDeep learning models in CV have helped tackle several specific EO use-cases such as \nscene classification~\\cite{LiuResNet2019},\nobject detection~\\cite{ZhangRCNN2019}, \nchange detection~\\cite{Mou2019RCNN_change_detection}, and \nsemantic segmentation~\\cite{YUAN2021Review_Semantic_Segm}, among others.\nHowever, the specificity of sensors require domain-specific models~\\cite{R3Net}.\nIn particular, the availability of several sensors to monitor areas \nhas motivated the activity of multi-modal sensor fusion which remains challenging for current models~\\cite{Hong2021MoreDiverseFusion}.\nMoreover, as the design of new vision-based methodologies can be time-consuming (trial and error),\nEO could benefit from \\emph{automated machine learning} (AutoML) algorithms.\n\nIn AutoML, NAS specializes in finding model configurations \nachieving optimal performances for a given dataset~\\cite{elsken2019neural,ojha2017review}.\nNAS methodologies have been proven to be powerful and efficient,\nwith strategies deriving from various families of optimization algorithm, e.g., \n\\emph{differentiable search}~\\cite{DARTS,traore2021esann}, \n\\emph{Bayesian optimization}~\\cite{camero2021bayesian}, \\emph{meta-heuristic}-based approaches~\\cite{stanley2002evolving,camero2020random,traore2021DataDrivenInit}.\nHowever, in practice, the difficulty of a NAS problem is hard to estimate, because the complexity of its components, namely the\nsearch space, search strategy, performance estimator and additional \\emph{tricks}, is hard to quantify~\\cite{elsken2019neural,Yang2020NasEvalHard}.\n\nThe fields of \\emph{evolutionary computation}, \\emph{optimization} and \\emph{complex systems} have long studied optimization processes and provide us with tools to analyze their behavior.\nIn particular, FLA~\\cite{pitzer2012comprehensive} aims at understanding \nand predicting performances of optimization algorithms.\nRecently, Traor\u00e9 et al. proposed the \\emph{Fitness Landscape Footprint}~\\cite{traore2021fitness}, \na framework to characterize NAS problems from the perspective of a search algorithm. \nThe following section introduces the \\emph{footprint}.\n\n\n\n\n\n\n\n\n\\section{A Framework for Comparative Fitness Landscape Analysis}\\label{sec:fitness-landscape}\n\nBefore describing the \\emph{footprint}, it is important to define what a fitness landscape is.\nLet~$S$ be the set of all possibles solutions of an optimization problem, i.e., the search space.\nLet~$f$ be the fitness function, which attributes to each candidate solution~$x \\in S$, \na fitness measurement~$f(x) \\in \\mathbb{R}$.\nLet~$N$ be a function providing a structure to the search space~$S$, the neighborhood relationship operator. \nThen, the fitness landscape~$\\mathcal{L} = (S, f, N)$ consists of combining the three above, in order \nto provide respectively with a set of possibles solutions, a function to evaluate them and another to interconnect them.\n\nGiven this definition, we are interested in a better understanding of a NAS optimization process. \nThe \\emph{footprint}~\\cite{traore2021fitness} serves this purpose \nof gaining insights into the process by describing its fitness landscape \nwith a set of eight (8) metrics measuring aspects such as\nthe distribution of fitness, ruggedness of the landscape or persistence of fitness. \nA~\\emph{footprint} includes: the mean and variance of fitness over~$S$, the ruggedness~$\\tau$, \nan enumeration of local optima, the positive and negative persistence and their area under the curve ($AuC$).\nThe following paragraphs describe these metrics.\n\nThe \\emph{fitness distance correlation} (FDC) is \noften interpreted a measure of the existence of \nsearch trajectories from randomly picked solutions to the known global optimum.\nIn practice, the FDC is not collected as a correlation score, \nbut visualized as the distribution of fitness versus \ndistance to the global optimum. \nIt writes as~$FDC(f,x^*, S)=\\{(d(x^*,y), f(y)) \\mid \\forall y \\in S\\}$, \nwhere~$S$ denotes the search space, \n$x^* \\in S$ is the global optimum, \n~$d$ a distance function. \n\nThe \\emph{ruggedness} of the landscape also helps assessing the difficulty of the process tackled.\nLet's consider a random walk~$RW$ in~$S$ of $N$ steps (models) and its corresponding fitness values.\nThe ruggedness~$\\tau$ consists in the auto-correlation length over ~$RW$ :~$\\tau=\\frac{1}{\\rho(1)}$, where ~$\\rho(k=1)$ is the serial-correlation coefficient for consecutive lags. \n\nIn \\cite{traore2021fitness}, the authors propose the metric of \\emph{persistence} characterizing the behavior of image classification models overtime. It measures the chances of solutions in the search space, \nto keep a rank~$N$ (top or bottom rank, based on fitness in test), as the training time grows.\nThis metric is complemented with its area under the curve ($AuC$), \nmeasuring the evolution of the persistence as $N$ grows.\n\nAnother way to characterize an optimization fitness landscape is to assess the existence of \\emph{local optima}.\nAs some search algorithms might get stuck in such sub-optimal areas of~$S$, \nan enumeration~\\cite{Hernando2012CardOptimaBenchmark} could help measure the difficulty of the search problem.\n\nLast but not least, the~\\emph{footprint} not only characterizes individual NAS landscapes, \nbut also enables the comparison of a handful considering potential changes \nin either components~$S$, ~$f$, or~$N$.\n\n\n\n\n\n\n\\section{Assessing and comparing the landscapes of NAS for various sensors}\\label{sec:footprint-for-fusion}\n\n\nThis study aims at investigating how the process of searching for neural architectures \nis affected by the type of sensor available as input.\nMore precisely, we seek to identify to what extent \ndoes searching with a given sensor \ndiffers from searching with another one.\nIn particular, we consider the case of a fixed search space, \ntraining pipeline (hyperparameters, duration, etc.)\nand evaluation protocol (fitness function). \nIn practice, we propose to tackle these questions by conducting \na comparative landscape study using the \\emph{footprint} (Section~\\ref{sec:fitness-landscape}).\n\nLet~$\\Sigma=\\{s_i, s_j, s_i + s_j\\}$ be the set of sensors available in our ML task.\nLet~$S$ be a search space of CNN image classifiers, each represented by a unique binary vector.\nConsidering this representation, we choose a neighborhood operator~$N(x)$ assigning to each solution of the search space, all the configurations that are one (1) \n\\textit{hamming distance} away from it. This operator~$N(x)$ writes as follows:~$N(x) = \\{ y \\in S \\mid d_{hamming(x, y)} = 1\\}$, \nwhere~$d_{hamming(x, y)}$ is the \\textit{hamming distance} between two solutions~$(x,y) \\in S^2$.\nAdditionally, we use as fitness function~$f_{s_{i}}$, the measurement of accuracy in test\nafter a training budget of~$b_t$, on an input sensor~$s_i$.\n\nSince we have access to various sensors~$s \\in \\Sigma$ for our ML task, \nthe fitness landscapes obtained write as follows:\n~$\\mathcal{L}_{s_i} = (S, f_{s_i}, N)$, ~$\\mathcal{L}_{s_j} = (S, f_{s_j}, N)$, ~$\\mathcal{L}_{s_k} = (S, f_{s_k}, N)$ for the sensor settings $(s_i, s_j, s_k) \\in \\Sigma^3$.\nIn particular, we consider the case of input level sensor fusion as:~$ s_k = s_i + s_j$.\nBesides, as noted above, the aim is a comparative study so we fix the search space~$S$ and the neighborhood operator~$N$ across all settings.\n\n\n\\section{Experimental Setup}\\label{sec:exp-setup}\nThis section introduces the NASBench-101 database, as well as a custom representation used to encode its solutions.\nThen, the So2Sat LCZ42 dataset used to evaluated solutions, followed by details on the evaluation protocol.\n\n\n\\subsection{NASBench-101}\n\nNASBench-101~\\cite{ying2019nasbench101} is a database containing a large pool of neural networks \nand their evaluations on the image classification dataset of CIFAR-10.\nIt aims at providing an exhaustive fitness measurement for all configurations (N=453k) \nin a search space of CNN image classifiers. \nThis search space defines a model configuration as an image classification backbone \nwith a head, body and tail.\nIts body consists of repeating three identical 'block' structures alternated with down-sampling modules.\nRegarding each block, it consist in a sequence of identical and elementary feed-forward units called cells.\nEach cell is represented by a directed DAG with a maximum number of nodes ($V \\leq 7$), \nmaximum number of edges ($E \\leq 9$) and a fixed listed of three (3) operators (Max-pool 3x3, Convolutional layer 1x1 and 3x3) \nlabelling each node. Therefore, a solution of the search space is identified by a cell, \nencoded in practice by both an adjacency matrix of variable size (upper triangular), and its list of operators.\nMoreover, The head of the model is a 3 x 3 convolution with 128 output channels, while the tail is a dense softmax layer.\n\n\\subsection{Custom feature representation}\n\nIn our experiments, \nwe construct a custom representation to enable solutions of the search space to be identified by a single vector.\nFirst, for our representation of the DAG, we do not label nodes.\nInstead, for the five (5) intermediate nodes out of seven (7) (one for IN and OUT), \nwe account for the fact that each could be one of three (3) operators.\nThus, the DAG contains exactly $N_{nodes}=(1+5*3+1) = 17$ nodes. \nThe new adjacency matrix is therefore of fixed length, i.e $L=17*17$ and non upper-triangular.\nFinally, we flatten the adjacency matrix to obtain a binary vector as identifier.\n\nRegarding the sampling of solutions $x \\in S$, we use the Latin Hypercube Sampling (LHS) for ensure fair data collection.\nBecause of a higher complexity of LHS on the large binary representation, we perform it on the intermediate \nrepresentation as a joint sampling of the original matrix and list of operations.\n\n\n\n\\subsection{So2Sat LCZ42}\n\n\n\nThe So2Sat LCZ42~\\cite{So2SatDataset} is a dataset of satellite imagery covering over forty-two (42) cities around the five (5) continents.\nIt provides with co-registered image patches from Sentinel-1 and Sentinel-2 sensors, \neach attributed with a single label.\nThis label associates an image to a class out of seventeen (17) possibilities, \nall representing the diverse Local Climate Zones (LCZ) around the globe.\nThe framework of LCZ proposes a generic way to describe the morphology \nof land use around the world in both urban and non-urban natural sites. \nThe classes are the following: Compact high-rise (1), Compact mid-rise (2), Compact low-rise (3), Open high-rise (4), \nOpen mid-rise (5), Open low-rise (6), Lightweight low-rise (7), Large low-rise (8), Sparsely built (9), and Heavy industry (10), \nDense trees (11), Scattered tree (12), Bush, scrub (13), Low plants (14), Bare rock or paved (15), Bare soil or sand (16), \nand Water (17).\nFigure~\\ref{fig:lcz42-dataset-visual} displays four (4) pairs of Sentinel-1 and Sentinel-2 image patches respectively from classes \n2, 4, 8 and 17.\n\nThe dataset comprises train ($N_t=352,366$), validation ($N_v=24,119$), and test ($N_{test}=24,188$) samples.\nThe training and validation samples originate from the same set of forty-two (42) cites, \nwhile those from the test set were collected in ten (10) additional cities.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\columnwidth]{Figures\/dataset\/description_lcz42_set2.png}\n \\caption{So2Sat LCZ42 samples. The top row contains SAR patches (Sentinel-1), followed by the associated Multi-spectral patches (Sentinel-2) in the bottom row.}\n \\label{fig:lcz42-dataset-visual}\n\\end{figure}\n\n\\subsection{Evaluation protocol}\nFor the purpose of training and evaluating on the same data distribution, \nwe use a custom setting consisting of training and testing sets made by randomly sampling respectively, \n$80\\%$ and $20\\%$ of the image patches of the original training-set. \nMoreover, as in \\cite{traore2021fitness} we speed-up the training procedure by only considering $P=35\\%$ of samples in the training set.\n\nAdditionally, we use the same search space~$S$ for all sensor settings.\nIn particular, we do not adapt the sampled models to use multiple sensors as input, \ninstead we do stack the data at an input level.\nWe trained ~$N=100$ randomly sampled models, once. After inspection and quality control, \nthere remain 100, 88 and 75 samples for Sentinel-1, 2 and both sensors.\nThe fitness is assessed in test using the Kappa-Cohen metric.\n\n\\section{Results}\\label{sec:results}\n\nThe following section presents results of comparison of search landscapes for various input sensors.\nFirst, we provide an analysis of distributions of fitness.\nThen, we show results of fitness distance correlation, followed by \nan analysis of random walks, as well as measurements of fitness persistence.\nLast but not least, we compare the \\emph{footprint} of the sensors.\n\n\\subsection{Density of Fitness}\\label{subsec:dos}\n\nFirst, we assess the ability of the search space in fitting the task with each sensor. \nFigure~\\ref{fig:pdf-36} and ~\\ref{fig:pdf-108e} display the probability density function (PDF) of fitness,\nrespectively after 36 and 108 epochs of training. \nThe first, second and third columns are, respectively, for using Sentinel-1, Sentinel-2 or both sensors as input.\n\nWe first take a look at the PDFs after 36 epochs of training.\nWhen using Sentinel-1, the distribution of fitness is wide and centered around low values ($\\mu=0.47, \\sigma=0.13$). \nSentinel-2 enables the distribution to improve by reaching a higher average and being more narrow ($\\mu=0.94, \\sigma=0.03$).\nUsing both sensors slightly worsens the fitness, providing with a lower mean and larger deviation ($\\mu=0.89, \\sigma=0.05$).\n\n\n\\begin{figure*}[ht]\n\\centering\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen1\/dos_test_acc_36e.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen2\/recollected\/dos_test_acc_36e.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen12\/dos_test_acc_36e.jpeg}\n \\caption{PDF of fitness after 36 epochs of training. \n From left to right are results using Sentinel-1, Sentinel-2 and both sensors as input.\n } \n \\label{fig:pdf-36}\n\\end{figure*}\n\n\n\\begin{figure*}[ht]\n\\centering\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen1\/dos_test_acc_108.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen2\/recollected\/dos_test_acc_108e.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen12\/dos_test_acc_108.jpeg}\n \\caption{PDF of fitness after 108 epochs of training. \n From left to right are results using Sentinel-1, Sentinel-2 and both sensors as input.\n } \n \\label{fig:pdf-108e}\n\\end{figure*}\n\nNext, we look at the PDFs after 108 epochs of training.\nOverall, the task is better handled in all sensor configurations.\nUsing Sentinel-1, the distribution improves by $17$ percentage points in mean fitness ($\\mu=0.64, \\sigma=0.13$).\nIn the case of Sentinel-2, most models fit well the data as the mean fitness increases and the deviation decreases ($\\mu=0.97, \\sigma=0.01$).\nWe observe similar results when using both sensors ($\\mu=0.94, \\sigma=0.04$).\n\n\n\n\n\\begin{figure*}[ht]\n\\centering\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen1\/pdf_fitting_36e.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen2\/recollected\/pdf_fitting_36e.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen12\/pdf_fitting_36e.jpeg}\n \\caption{Fitting PDF to fitness with various theoretical functions after 36 epochs of training. \n From left to right are results using Sentinel-1, Sentinel-2 and both sensors. \n The red, blue and green colors correspond to Beta, Log-normal and Weibull distributions, respectively.\n } \n \\label{fig:pdf-fitting-36e}\n\\end{figure*} \n\n\n\\begin{figure*}[ht]\n\\centering\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen1\/emp_vs_theoretical_CDFS_36e.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen2\/recollected\/cdf_36.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen12\/emp_vs_theoretical_CDFS_36e.jpeg}\n \\caption{ Cumulative PDFs of fitness versus theoretical density functions, after 36 epochs of training. \n The plot uses the same color and sensor convention as Figure~\\ref{fig:pdf-fitting-36e}.\n \n } \n \\label{fig:cdf-36e}\n\\end{figure*} \n\n\nBesides, we seek to identify for each sensor, \nif the behavior of the search space follows a specific theoretical distribution. \nFor this, we consider the more challenging scenario \nof selecting models after only 36 epochs of training.\n\nFigure~\\ref{fig:pdf-fitting-36e} and ~\\ref{fig:cdf-36e} show results of fitting empirical distributions of fitness,\nwith various theoretical distributions.\nFigure~\\ref{fig:pdf-fitting-36e} and ~\\ref{fig:cdf-36e} show, respectively, PDFs and cumulative density functions (CDF), \nall fitted with the Beta (red), Weibull (green) and Log-normal (blue) distributions.\nSimilarly, the first, second and third columns are respectively, \nfor using Sentinel-1, Sentinel-2 or both sensors as input.\nTo complement the plots, Table~\\ref{tab:pdf-fitting-error-s1},~\\ref{tab:pdf-fitting-error-s2} and~\\ref{tab:pdf-fitting-error-s12} provide with the respective fitting errors.\n\n\n\n\\begin{table}[h!]\n \\centering \n\\scriptsize\n\\begin{tabular}{|l|l|l|l|l|}\\hline \n\\textit{Error Metric \/ Function} &\\textit{Beta} & \\textit{Weibull} & \\textit{Log-normal}\\\\\n\\hline\nLikelihood & 46.28 & 44.52 & \\textbf{53.63} \\\\ \n\\hline\nAIC & -88.56 & -85.04 & \\textbf{-103.26} \\\\ \n\\hline\nBIC &-83.92 & -80.41 & \\textbf{-98.62}\\\\ \n\\hline\n\\end{tabular}\n\\caption{PDF fitting error for Sentinel-1 } \n\\label{tab:pdf-fitting-error-s1}\n\\end{table}\n\n\\vspace{0.2cm}\n\n\\begin{table}[h!]\n \\centering \n\\scriptsize\n\\begin{tabular}{|l|l|l|l|l|}\\hline \n\\textit{Error Metric \/ Function} &\\textit{Beta} & \\textit{Weibull} & \\textit{Log-normal}\\\\\n\\hline\nLikelihood & \\textbf{165.18} & 164.70 & 155.05 \\\\ \n\\hline\nAIC & \\textbf{-326.36} & -325.40 & -306.10 \\\\ \n\\hline\nBIC & \\textbf{-321.80} & -320.84 & -301.54\\\\ \n\\hline\n\\end{tabular}\n\\caption{PDF fitting error for Sentinel-2 } \n\\label{tab:pdf-fitting-error-s2}\n\\end{table}\n\n\\vspace{0.2cm}\n\n\\begin{table}[h!]\n \\centering \n\\scriptsize\n\\begin{tabular}{|l|l|l|l|l|}\\hline \n\\textit{Error Metric \/ Function} &\\textit{Beta} & \\textit{Weibull} & \\textit{Log-normal}\\\\\n\\hline\nLikelihood & \\textbf{109.72} & 109.04 & 107.78\\\\ \n\\hline\nAIC & \\textbf{-215.44} & -214.08 & -211.57 \\\\ \n\\hline\nBIC & \\textbf{-211.09} & -209.74 & -207.22\\\\ \n\\hline\n\\end{tabular}\n\\caption{PDF fitting error for both sensors as input } \n\\label{tab:pdf-fitting-error-s12}\n\\end{table}\n\nOverall, the empirical distributions are closely fitted with the selected theoretical distributions.\nFor Sentinel-1, the best candidate is the Log-normal with the largest fitting likelihood, and lowest AIC and BIC error scores. When using Sentinel-2 or both sensors, Beta matches the best the empirical distributions. \n\nTo summarize, \nthe capacity of the search space in fitting the task with the Sentinel-1 sensor appears limited from the PDF perspective.\nIndeed, despite longer training time (108 epochs) the fitness distribution remains far worse than using Sentinel-2.\nUsing Sentinel-2 only, the task can be fitted well enough in particular given long training time.\nCombining Sentinel-1 to Sentinel-2 worsens the distribution of fitness (lower mean, larger deviation).\nTherefore, there is no tangible benefits in fitness, from sensor fusion using the current search space.\nMoreover, results indicate the feasibility in modeling the empirical distributions of fitness for each sensor.\n\n\\subsection{Fitness Distance Correlation}\\label{subsec:fdc}\n\nNext, we analyse the fitness landscape for the various sensor configurations.\nFigure~\\ref{fig:profiles-36} and \\ref{fig:profiles-108} show results of FDC for the three (3) input sensor settings.\nThe layout of the plots follows the convention of Figure~\\ref{fig:pdf-36}.\n\n\n\\begin{figure*}[ht]\n\\centering\n \\includegraphics[width=0.3\\textwidth]{Figures\/sen1\/profile_36e.jpeg}\n \\includegraphics[width=0.3\\textwidth]{Figures\/sen2\/recollected\/profile_36e.jpeg}\n \\includegraphics[width=0.3\\textwidth]{Figures\/sen12\/profile_36e.jpeg}\n \\caption{Fitness distance correlation after 36 epochs of training.\n From left to right are shown results using Sentinel-1, Sentinel-2 and both sensors. \n } \n \\label{fig:profiles-36}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n\\centering\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen1\/profile_108e.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen2\/recollected\/profile_108e.jpeg}\n \\includegraphics[width=0.275\\textwidth]{Figures\/sen12\/profile_108e.jpeg}\n \\caption{Fitness distance correlation after 108 epochs of training.\n From left to right are shown results using Sentinel-1, Sentinel-2 and both sensors. \n } \n \\label{fig:profiles-108}\n\\end{figure*}\n\nFirst, we consider the FDC after 36 epochs of training (see Figure~\\ref{fig:profiles-36}).\nOverall, for all sensors, we observe that the respective landscapes are rather rough.\nIndeed, the distribution of fitness per hamming distance to the optimum are relatively wide.\nFor instance, when using Sentinel-1 we notice that solutions at the \\textit{Hamming-distances} $d_{hamming}=\\{8, 11, 13\\}$ display up to 35\\% percentage point in fitness difference.\nAlso, We notice a landscape around low fitness values (c.a 47\\%).\nUsing Sentinel-2 provides with a landscape centered at much higher values (c.a 94\\%).\nWe also notice a consistent increase in fitness, \nas the hamming distance to the optimum decreases.\nSimilarly, the landscape associated with using both sensors is of high fitness. \nHowever, the slope of gained fitness per travelled distance to the optimum worsens (less consistent), compared to the one obtained with Sentinel-2 as input.\n\nThen, we consider the FDC after 108 epochs of training (see Figure~\\ref{fig:profiles-108}).\nOverall, the landscape tends to be more flat and with an increased fitness.\nIn particular for Sentinel-2 or both sensors as input, \nthe flatness is indicated by more narrow distribution of fitness \nat the various distances to the optimum.\nThis also is complemented by potential search trajectories \nthat have little improvements in fitness per travelled distance to the global optimum. \nUsing both sensors brings us a similar behaviors, \nexcept for the existence of a set of models providing poorer fitness values, \nall located at $d_{hamming}=7$ from the optimum.\nThe case of Sentinel-1 is rather odd as there appears to be a favorable (negative) slope, as if the landscape had not converged.\n\nTo summarize, the fitness landscape of So2Sat LCZ42 is rougher when training is limited (36 epochs), \nand flatter towards higher fitness when training long enough (108 epochs) solutions in the search space.\nAs observed when analyzing distributions of fitness (Figure~\\ref{fig:pdf-36}, \\ref{fig:pdf-108e}), \nthis NAS problem benefits better from using Sentinel-2 as input, \nwith improvements in slope and overall fitness in its landscape.\nTherefore, these results complement the analysis of PDFs \nby showing benefits in performances, this time from the perspective of potential NAS algorithm trajectories. It also shows that with the current search space, the search behaviour is worse when using both sensors as input.\n\n\n\\subsection{Random Walk Analysis}\\label{subsec:random-walks}\n\n\nFurthermore, we investigate the behavior \nof local search-based algorithm depending on the input sensor.\nMore precisely, this is done by analyzing random walks.\n\n\\begin{figure\n\\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/random_walks\/rw22_various_sensors_36e.png}\n \\caption{A random walk evaluated with the four (4) different sensor settings. \n } \n \\label{fig:random-walks}\n\\end{figure}\n\n\nFigure~\\ref{fig:random-walks} displays the route of a random walk \nevaluated on for four (4) different sensor settings.\nThe walk itself consists of one hundred (100) steps in the search space.\nAt each step, the selected model is evaluated after being trained for 36 epochs. \nThe blue, yellow, green and red curves are, respectively, \nfor evaluating the fitness with Sentinel-1, Sentinel-2, both sensors, or CIFAR-10 as input. \nAll curves were smoothed with a moving average of five (5) steps.\nWe also consider CIFAR-10, since its fitness evaluations were freely \navailable (ground-truth in NAS-Bench-101) and could serve as reference for comparison, and trouble-shooting.\n\n\nThe evaluation of the walk on Sentinel-1 provides with the lowest overall fitness (~$\\mu=0.44$) and the most rugged route (~$\\tau=17.84$).\nOn the other hand, using Sentinel-2 or both sensors together, provides with more smooth paths, at much higher values. Indeed, the respective averages of fitness are ~$\\mu=0.93$ and ~$\\mu=0.89$. \nThe ruggedness values are ~$\\tau=1.54$ and ~$\\tau=6.03$. \nAlso , the curvature of both routes \\textit{visually} look alike.\nRegarding CIFAR-10, we observe intermediate fitness and ruggedness (~$\\mu=0.64$ and ~$\\tau=1.56$).\nThe relatively large amplitude in fitness and similar curvature, despite lower ruggedness, makes its route look more similar to the one evaluated with Sentinel-1.\n\nAs observed when analyzing FDCs (Section~\\ref{subsec:fdc}),\nSentinel-1 provides with poorer trajectories (lower fitness, larger ruggedness), \nsuggesting either a sensor being unsuitable for the task \nor a search space~$S$ not suitable for the sensor.\nSimilar results (curvature, lower fitness) obtained for ground-truth evaluations on CIFAR-10 suggest \nthat a higher ruggedness seems to associate with harder tasks and lower convergence of models in a random walk route.\n\nTo summarize, the use of either Sentinel-2 or both sensors after only 36 epochs\nenables a NAS route to be of higher smoothness and fitness.\n\n\\subsection{Persistence}\\label{subsec:persistence}\n\nNext, we study the behaviour of solutions in the search space, \nfrom the perspective of persistence in their ranking.\n\nFigure~\\ref{fig:persistence-positive} and \\ref{fig:persistence-negative} \nshow measurements of positive and negative persistence.\nWe consider samples collected for experiments related to section ~\\ref{subsec:dos} and \\ref{subsec:fdc}.\nFor each setting, the blue curve represents the reference population: the models at a given~$Rank-N$\nbased on their fitness after 4 epochs of training.\nThe yellow curve display the share of these models maintaining the same~$Rank-N$ after 12 epochs.\nThe green and red show the same (intersection of sets) respectively after 36 and 108 epochs of training.\nThe positive and negative Persistence refer to using the top and bottom ~$N$ rank function (Nth percentile).\n\n\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width=0.4\\textwidth]{Figures\/persistence\/positive_sentinel_1_quantile_based.png}\n \\qquad\n \\includegraphics[width=0.4\\textwidth]{Figures\/persistence\/positive_sen2_quantile_other_label.png}\n \\qquad\n \\includegraphics[width=0.4\\textwidth]{Figures\/persistence\/positive_both_sentinel_quantile_based.png}\n \\caption{Positive persistence for Sentinel-1, Sentinel-2 and when using both sensors.} \n \\label{fig:persistence-positive}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width=0.4\\textwidth]{Figures\/persistence\/negative_sentinel_1_quantile_based.png}\n \\qquad\n \\includegraphics[width=0.4\\textwidth]{Figures\/persistence\/negative_sen2_quantile_other_label.png}\n \\qquad\n \\includegraphics[width=0.4\\textwidth]{Figures\/persistence\/negative_both_sentinel_quantile_based.png}\n \\caption{Negative persistence for Sentinel-1, Sentinel-2 and when using both sensors.} \n \\label{fig:persistence-negative}\n\\end{figure}\n\nFirst we have a look at the positive persistence (see Figure~\\ref{fig:persistence-positive}).\nOverall, we observe that the larger the fitness a sensor can provide (see Figure~\\ref{fig:pdf-36} \\ref{fig:pdf-108e}), \nthe larger its persistence (N$<$25) across all training budgets.\nMore precisely in terms of~\\textit{Area under the Curve (N$<$25)}, Sentinel-2 ($AuC=0.14$) improves over the use of both sensors ($AuC=0.07$), which also improves over a single Sentinel-1 sensor ($AuC=0.04$).\n\n\nNext we take a look at the negative persistence (see Figure~\\ref{fig:persistence-negative}).\nOverall, Sentinel-2 generates the larger persistence ($P=31.82$, $AuC=0.26$), while the use of Sentinel-1 ($P=4.0$, $AuC=0.01$) or both sensors \n($P=5.26$, $AuC=0.01$) result in average measurements.\n \nTo summarize, we observe \nsimilar trend across sensors for both positive and negative persistence.\nA larger fitting capacity results in a larger persistence (Sentinel-2).\nIn the case of Sentinel-1, the limited ability to fit the sensor \nmight hinder the ability of models to keep their ranking (potential instabilities during training). \nIn turn, this might result in a poorer persistence. \nFor the better fitted sensor (Sentinel-2) chances of finding top-25\\% and bottom-25\\% performers are considerable ($P=13.7$, $P=31.82$). \n\n\n\\subsection{Fitness Landscape Footprint}\\label{subsec:footprint}\nThe results obtained in the previous sections are summarized by the \\textit{footprint} for each data source.\nFigure~\\ref{fig:footprnts} displays the \\textit{footprint} for Sentinel-1 (blue), Sentinel-2 (yellow) and both sensors together (green).\nThis is done using considering only 36 epochs of training.\n\nAs observed in section~\\ref{subsec:dos}, the search space appears better suited to fit Sentinel-2, than Sentinel-1. \nIndeed, Sentinel-2 enables reaching a larger mean fitness ($\\mu=0.94$) and lower standard deviation ($\\sigma=0.03$) than the other sensors.\nPerforming an input-level fusion slightly worsens the fitness ($\\mu=0.89, \\sigma=0.05$).\nSimilarly, the search landscape of Sentinel-2 appears more favourable to search trajectories. \nThis transpires through random walk routes that are smoother and of higher fitness ($\\mu=0.93$, $\\tau=1.54$).\nRegarding the evolution in the ranking of samples, \na larger fitness results in a larger positive and negative persistence.\nIndeed, this transpires in the larger measurements obtained for Sentinel-2 (\\textit{Pos. AuC=0.14, Neg. AuC=0.26}). \n\n\n\\section{Conclusion}\\label{sec:conclusion}\n\nIn this study, we investigate the impact of the choice of an input sensor \non the performance of a neural architecture search strategy.\nMore precisely, we want to know to what extent does searching with a given sensor differs from searching with other sensors, \nin the context of neural architecture optimization.\nAre there benefits or drawbacks in searching with fused sensors ?\n\nTo answer such questions, we use the framework of ~\\textit{Fitness Landscape Footprint}, \nthat we apply on the Real World image classification task So2Sat LCZ42, \nand analyze the process of searching CNN image classifiers provided in the NASBench-101 database.\nAfter sampling and evaluating solutions with three different sensor settings (including input level fusion), \nwe provide a comparative analysis assessing for instance the distribution of fitness, fitness distance correlation and ruggedness of the landscapes.\n\nOverall, we observe a consistent improvement in the capacity of fitting all sensors, the longer the training time.\nUsing Sentinel-2 enables larger fitness, over Sentinel-1 or an input-level fusion of both sensors.\nSimilar results are observed when analysing search landscapes. Indeed, the longer the training, \nthe landscape evolve from high ruggedness to flatness. \nMoreover, search strategies might benefit from a deployment on Sentinel-2, \nas it provides with routes that are smoother, of higher fitness, higher gain per distance travelled, \nand higher persistence in ranking of models. \nWhen a sensor can be fit well enough (Sentinel-2, fusion), we observe very similar behaviour in terms of trajectories (smoothness, ruggedness, fitness). \nThis strongly indicates that search trajectories associated to different sensors are comparable \nwhen the search space is able to fit them decently enough. \n\nAs future work, we propose to investigate how to use the gained insights to help build speed-up techniques for NAS strategies.\nSuch technique could rely, for instance, on searching with a sensor (or a subset of given sensors),\nhelping approximate the search with a more expensive to evaluate target sensor. \n \n\n\\section*{ACKNOWLEDGEMENTS}\\label{ACKNOWLEDGEMENTS}\nAuthors acknowledge support by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. [ERC-2016-StG-714087], Acronym: \\textit{So2Sat}), by the Helmholtz Association\nthrough the Framework of Helmholtz AI [grant number: ZT-I-PF-5-01] - Local Unit ``Munich Unit @Aeronautics, Space and Transport (MASTr)'' and Helmholtz Excellent Professorship ``Data Science in Earth Observation - Big Data Fusion for Urban Research'' (W2-W3-100), by the German Federal Ministry of Education and Research (BMBF) in the framework of the international future AI lab \"AI4EO -- Artificial Intelligence for Earth Observation: Reasoning, Uncertainties, Ethics and Beyond\" (Grant number: 01DD20001) and the grant DeToL.\n\n{\n\t\\begin{spacing}{1.17}\n\t\t\\normalsize\n\t\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}