diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzetrp" "b/data_all_eng_slimpj/shuffled/split2/finalzzetrp" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzetrp" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction and main results}\nWe start with some notation and terminology. Let\n$(\\R^d,\\<\\cdot,\\cdot\\>,|\\cdot|)$ be the $d$-dimensional Euclidean\nspace, and $\\R^d\\otimes\\R^m$ the collection of all $d\\times m$\nmatrices endowed with the Hilbert-Schmidt norm $\\|\\cdot\\|$. For\nfixed $r_0>0$, $\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r:=C([-r_0,0];\\R^d)$ stands for the family of all\ncontinuous functions $f:[-r_0,0]\\rightarrow\\R^d$ which is a Banach space\nwith the uniform norm $\\|f\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle:=\\sup_{-r_0\\le v\\le 0}|f(v)|$. Given any\ninteger $ p\\ge1$, we use $\\Theta$ to denote a bounded, open and convex\nsubset of $\\R^p$ whose closure is written as $\\bar\\Theta$. Let $\\mathcal\n{P}(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$ be the totality of all probability measures on $\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$. Set\n$\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r):=\\{\\mu\\in\\mathcal\n{P}(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r):\\mu(\\|\\cdot\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2):=\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\\|\\xi\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\mu(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\xi)<\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\}$.\n$(\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r),\\W_2)$ is a Polish space under the Warsserstein\ndistance $\\mathbb{W}_2$ on $\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$ defined by\n$$\\W_2(\\mu,\\nu):= \\inf_{\\pi\\in \\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r(\\mu,\\nu)} \\bigg(\\int_{\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\\times\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r} \\|\\xi-\\eta\\|^2_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\pi(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D \\xi,\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D \\eta)\\bigg)^{\\ff 1 2},\\ \\ \\mu,\\nu\\in \\mathcal\n{P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r),$$ where $\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r(\\mu,\\nu)$ is the set of couplings for $\\mu$\nand $\\nu$. As usual, we use $\\lfloor a\\rfloor$ to denote the integer part of $a\\ge0.$\n\n\n\n\n\n\n\n\n\n\n\nThe time evolution for most of stochastic dynamical systems depends\nnot only on the present state but also on the past path. So,\npath-dependent (i.e., functional) SDEs are much more desirable; see,\ne.g., the monograph \\cite{M84}. Since the pioneer work \\cite{IN} due\nto It\\^o and Nisio, path-dependent SDEs have been investigated\nconsiderably owing to their theoretical and practical importance;\nsee, e.g., Hairer et al. \\cite{HMS11}, Wang \\cite{W18} and the\nreferences within.\n\n McKean-Vlasov SDEs, which are SDEs with coefficients dependent on the\n law, were initiated by \\cite{Mc} inspired by Kac's programme in Kinetic\n theory. An excellent and thorough account of the general theory of McKean-Vlasov\n SDEs and their particle approximations can be found in \\cite{sz}. McKean-Vlasov\n SDEs\nare alternatively referred to as mean-field SDEs in the literature, which have\nwide applications in interacting particle systems, optimal control problems,\ndifferential games, just to mention but a few. Recently, McKean-Vlasov SDEs have been\nextensively investigated on, e.g., wellposedness of strong\/weak solutions (cf.\n\\cite{DST,HSS,LMb,MV,W16}), Freidlin-Wentzell large deviation principles (cf. \\cite{DST}), ergodicity (cf. \\cite{B14,EGZ,Ve}), links with nonlinear partial differential\nequations (cf. \\cite{BLPR,HRW,HRW}), and distribution properties (cf.\\cite{Huang,W18}).\n\nOn the other hand, from stochastic and\/or statistical aspects, there exist\nunknown parameters in various type SDEs arising in mathematical modeling (cf.\n\\cite{B08}). Hence, there are vast of investigations paying attention to\nparameter estimations for SDEs via maximum likelihood estimator,\nleast squares estimator (LSE for short), trajectory-fitting estimator, among others.\nSee, for instance, \\cite{Ku04,LS01,M05,P199,SY06}. In the same vein, the parameter estimations for SDEs (without path-dependence) with small noises have been developed very\nwell; see, e.g., \\cite{GS,HL,Long10,L09,LMS,LSS,M10,SM03,U04,U08}, and references\ntherein.\n\nFrom above discussion, it is very natural to consider SDEs together with all four features of path dependence, distribution dependence, small noises and unknown parameter. So,\nin the present work, we focus on the following path-distribution SDE\n\\begin{equation}\\label{eq1}\n\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D X^\\vv(t)=b(X_t^\\vv,\\mathscr{L}_{X_t^\\vv},\\theta)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt+\\vv\\,\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^\\vv,\\mathscr{L}_{X_t^\\vv})\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(t),\n~~~t>0,~~~~X_0^\\vv=\\xi\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r.\n\\end{equation}\nHerein, $\\vv\\in(0,1)$ is the scale parameter; for fixed $t$,\n$X_t^\\vv(v):=X^\\vv(t+v), v\\in[-r_0,0],$ is called the segment (or\nwindow) process generated by $X^\\vv(t)$; $\\mathscr{L}_{X_t^\\vv}$\nstands for the distribution of $X_t^\\vv$; $b:\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\\times\\mathcal\n{P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)\\times\\Theta\\rightarrow\\R^d$ and $\\sigma} \\def\\ess{\\text{\\rm{ess}}:\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\\times\\mathcal\n{P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)\\rightarrow\\R^d\\otimes\\R^m$ are continuous w.r.t. the\nfirst variable and the second variable;\n $\\Theta\\ni\\theta $ is an unknown parameter whose true value is\n written as\n$\\theta_0\\in\\Theta$; and $(B(t))_{t\\ge0}$ is an $m$-dimensional\nBrownian motion on a filtered probability space\n$(\\OO,\\F,(\\F_t)_{t\\ge0},\\P)$ satisfying the usual conditions, that\nis, $\\F_t$ is non-decreasing (i.e., $\\F_s\\subseteq\\F_t, s\\le t),$\n$\\F_0$ contains all $\\P$-null sets and $\\F_t$ is right continuous\n(i.e., $\\F_t=\\F_{t+}:=\\bigcap_{s\\uparrow t}\\F_s$).\n\n\nTo guarantee the existence and uniqueness of solutions to \\eqref{eq1},\nwe assume that, for any $\\zeta_1,\\zeta_2\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$, $\\mu,\\nu\\in\\mathcal\n{P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$, and $\\theta\\in\\Theta,$\n\\begin{enumerate}\n\\item[({\\bf A1})] There exist\n$\\aa_1,\\aa_2>0$ such that\n\\begin{equation*}\n\\<\\zeta_1(0)-\\zeta_2(0),b(\\zeta_1,\\mu,\\theta)-b(\\zeta_2,\\nu,\\theta)\\>\\le\\aa_1\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\aa_2\\mathbb{W}_2(\\mu,\\nu)^2;\n\\end{equation*}\n\n\n\\item[({\\bf A2})] There exist $\\bb_1,\\bb_2>0$ such that\n\\begin{equation*}\n\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta_1,\\mu)-\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta_2,\\nu)\\|^2\\le\n\\bb_1\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\bb_2\\mathbb{W}_2(\\mu,\\nu)^2.\n\\end{equation*}\n\\end{enumerate}\n\nFrom \\cite[Theorem 3.1]{HRW}, \\eqref{eq1} has a unique\nstrong solution $(X^\\vv(t))_{t\\ge-r_0}$ under the assumptions ({\\bf\nA1}) and ({\\bf A2}). For any $\\zeta_1,\\zeta_2\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$,\n$\\mu,\\nu\\in\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$, and $\\theta\\in\\Theta,$ if there\nexist $\\aa,\\bb>0$ such that\n\\begin{equation*}\n\\<\\zeta_1(0)-\\zeta_2(0),b(\\zeta_1,\\mu,\\theta)-b(\\zeta_2,\\mu,\\theta)\\>\\le\\aa\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\n\\end{equation*}\nand\n\\begin{equation*}\n|b(\\zeta_2,\\mu,\\theta)-b(\\zeta_2,\\nu,\\theta)|\\le\n\\bb\\mathbb{W}_2(\\mu,\\nu),\n\\end{equation*}\nthen ({\\bf A1}) holds.\n\n\nWithout loss of generality, we arbitrarily fix the time horizontal $T>0$ and assume that there exist positive integers $n,M$ sufficiently large such that\n$\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho:=\\frac{T}{n}=\\ff{r_0}{M}$. Now we define the continuous-time\ntamed Euler-Maruyama (EM) scheme (see, e.g., \\cite{HJK}) associated\nwith \\eqref{eq1}\n\\begin{equation}\\label{q3}\n\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D Y^\\vv(t)=b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t+ \\vv\\,\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(t),~~~~t>0\n\\end{equation}\nwith the initial value $Y^\\vv(t)=X^\\vv(t)=\\xi(t)$ for any $\nt\\in[-r_0,0]$, where\n\\begin{itemize}\n\\item $t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho:=\\lfloor t\/\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rfloor\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho$ for $t\\ge0;$\n\n\\item For any $\\zeta\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$ and $\\mu\\in\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$,\n\\begin{equation}\\label{e4}\nb^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta,\\mu,\\theta):=\\ff{b(\\zeta,\\mu,\\theta)}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta,\\mu,\\theta)|},~~~~\\aa\\in(0,1\/2];\n\\end{equation}\n\n\\item For $k=0,1,\\cdots,n,$\n$\\bar Y_{k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv=\\{ \\bar Y_{k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv(s):-r_0\\le s\\le0\\}$, a\n$\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$-valued random variable, is defined by\n\\begin{equation}\\label{w2}\n\\bar\nY_{k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv(s)=Y^\\vv((k-i)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\ff{s+i\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\{Y^\\vv((k-i)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\n-Y^\\vv((k-i-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\}\n\\end{equation}\nfor any $s\\in[-(i+1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho,-i\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho]$, $i=0,1,\\cdots,M-1$, that is, $\\bar\nY_{k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv$ is the linear interpolation of the points\n$(Y^\\vv(l\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho))_{l=k-M,\\cdots,k}$.\n\\end{itemize}\n\nWe denote $(Y_t^\\vv)_{t\\ge0}$ by the segment process generated by\n$(Y^\\vv(t))_{t\\ge-r_0}$. It is worthy to point out that $\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$ is defined by \\eqref{w2} rather than by $\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}(s)=\\bar Y^\\vv(t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho+s)$ for any $s\\in[-r_0,0]$\n Based on the\ncontinuous-time tamed EM algorithm \\eqref{q3}, we design the\nfollowing contrast function\n\\begin{equation}\\label{eq2}\n\\Psi_{n,\\vv}(\\theta)=\\vv^{-2}\\delta^{-1}\\sum_{k=1}^nP_k^*(\\theta)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta),\n\\end{equation}\nin which, for $k=1,\\cdots,n$,\n\\begin{equation}\\label{w1}\nP_k(\\theta):=Y^\\vv(k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)-Y^\\vv((k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)-b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho,~\n\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv):=(\\sigma} \\def\\ess{\\text{\\rm{ess}}\\si^*)^{-1}(\\bar\nY_{k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv}).\n\\end{equation}\nFor more motivations on the construction of constrast function\nabove, we refer to Ren-Wu \\cite{RW}. To obtain the LSE of\n$\\theta\\in\\Theta$, it is sufficient to choose an element\n$\\hat\\theta_{n,\\vv}\\in\\Theta$ satisfying\n\\begin{equation*}\n\\Psi_{n,\\vv}(\\hat\\theta_{n,\\vv})=\\min_{\\theta\\in\\Theta}\\Psi_{n,\\vv}(\\theta).\n\\end{equation*}\nWhence, for\n\\begin{equation*}\n\\Phi_{n,\\vv}(\\theta):=\\vv^2(\\Psi_{n,\\vv}(\\theta)-\\Psi_{n,\\vv}(\\theta_0)),\n\\end{equation*}\none has\n\\begin{equation}\\label{eq4}\n\\Phi_{n,\\vv}(\\hat\\theta_{n,\\vv})=\\min_{\\theta\\in\\Theta}\\Phi_{n,\\vv}(\\theta).\n\\end{equation}\nWe shall rewrite $\\hat\\theta_{n,\\vv}\\in\\Theta$ such that \\eqref{eq4}\nholds true as\n\\begin{equation*}\n\\hat\\theta_{n,\\vv}=\\arg\\min_{\\theta\\in\\Theta}\\Phi_{n,\\vv}(\\theta),\n\\end{equation*}\nwhich is called the LSE of the unknown parameter\n$\\theta\\in\\Theta$.\n\nTo discuss the consistency of LSE (see Theorem \\ref{th1} below), we\nfurther suppose that, for any $\\zeta,\\zeta_1,\\zeta_2\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$,\n$\\mu,\\nu\\in\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$, and $\\theta\\in\\Theta,$\n\\begin{enumerate}\n\n\\item[({\\bf B1})] There exist $q_1,L_1>0$ such that\n\\begin{equation*}\n|b(\\zeta_1,\\mu,\\theta)-b(\\zeta_2,\\nu,\\theta)|\\le\nL_1\\Big\\{(1+\\|\\zeta_1\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_1}+\\|\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_1})\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mu,\\nu)\\Big\\};\n\\end{equation*}\n\\item[({\\bf B2})] There exist $q_2,L_2>0$ such that\n\\begin{equation*}\n\\sup_{\\theta\\in\\bar\\Theta}\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)(\\zeta_1,\\mu,\\theta)-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)(\\zeta_2,\\nu,\\theta)\\|\\le\nL_2\\Big\\{(1+\\|\\zeta_1\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_2}+\\|\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_2})\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mu,\\nu)\\Big\\},\n\\end{equation*}\nwhere $(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)$ is the gradient operator w.r.t. the third\nspatial variable;\n\n\\item[({\\bf B3})] $(\\sigma} \\def\\ess{\\text{\\rm{ess}}\\si^*)(\\zeta,\\mu)$ is invertible, and\nthere exist $q_3,L_3>0$ such that\n\\begin{equation*}\n\\|(\\sigma} \\def\\ess{\\text{\\rm{ess}}\\si^*)^{-1}(\\zeta_1,\\mu)-(\\sigma} \\def\\ess{\\text{\\rm{ess}}\\si^*)^{-1}(\\zeta_2,\\nu)\\|\\le\nL_3\\Big\\{(1+\\|\\zeta_1\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_3}+\\|\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_3})\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mu,\\nu)\\Big\\};\n\\end{equation*}\n\\item[({\\bf B4})] There exists a constant $K>0$ such that\n\\begin{equation*}\n|\\xi(t)-\\xi(s)|\\le K|t-s|,~~~t,s\\in[-r_0,0],\n\\end{equation*}\nwhere $\\xi(\\cdot)$ stands for the initial value of \\eqref{eq1}.\n\\end{enumerate}\n\nIn order to reveal the asymptotic distribution of LSE (see Theorem\n\\ref{th2} below), we in addition assume that\n\\begin{enumerate}\n\\item[({\\bf C})] There exist $q_4,L_4>0$ such that\n\\begin{equation*}\n\\begin{split}\n&\\sup_{\\theta\\in\\bar\\Theta}\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb^*))(\\zeta_1,\\mu,\\theta)-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb^*))(\\zeta_2,\\nu,\\theta)\\|\\\\\n&\\le\nL_4\\Big\\{(1+\\|\\zeta_1\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_4}+\\|\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_4})\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mu,\\nu)\\Big\\},\n\\end{split}\n\\end{equation*}\nwhere $b^*$ means the transpose of $b.$\n\\end{enumerate}\n\n\n\nNext we consider the following deterministic path-dependent ordinary\nequation\n\\begin{equation}\\label{k1}\n\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D X^0(t)=b(X_t^0,\\mathscr{L}_{X_t^0},\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt,~~~t>0,~~~X_0^0=\\xi\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r.\n\\end{equation}\nUnder the assumption ({\\bf A1}), \\eqref{k1} is wellposed. In\n\\eqref{k1}, $\\mathscr{L}_{X_t^0}$ is indeed a Dirac's delta measure\nat the point $X_t^0$ as $X_t^0$ is deterministic. To unify the\nnotation, we keep the notation $\\mathscr{L}_{X_t^0}$ in lieu of\n$\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{X_t^0}$. We remark that ({\\bf B4}) is imposed to guarantee\nthat the linear interpolation $\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}$ tends to $X_t^0$\nin the moment sense, see Lemma \\ref{le1} below.\n\n\n\n\n For any random variable $\\zeta\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$ with\n$\\mathscr{L}_\\zeta\\in\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$, set\n\\begin{equation}\\label{p1}\n\\Gamma(\\zeta,\\theta,\\theta_0):=b(\\zeta,\\mathscr{L}_\\zeta,\\theta_0)\n-b(\\zeta,\\mathscr{L}_\\zeta,\\theta),~~\n~~\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta,\\theta,\\theta_0):=b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta,\\mathscr{L}_\\zeta,\\theta_0)\n-b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta,\\mathscr{L}_\\zeta,\\theta),\n\\end{equation}\nand, for any $\\theta\\in\\Theta,$\n\\begin{equation*}\\label{e0}\n\\Xi(\\theta)=\\int_0^T\\Gamma^*(X_t^0,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\Gamma(X_t^0,\\theta,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt,\n\\end{equation*}\nwhere $(X_t^0)_{t\\ge0}$ is the functional solution to \\eqref{k1}.\n\nThe theorem below is concerned with the consistency of the LSE for\nthe parameter $\\theta\\in\\Theta$, which is the first contribution of\nour work.\n\n\n\\begin{thm}\\label{th1}\n Let $({\\bf A1})-({\\bf A2})$ and $({\\bf B1})-({\\bf B4})$ hold and\nassume further that $\\Xi(\\theta)>0$ for $\\theta\\neq\\theta_0$. Then\n\\begin{equation*}\n\\hat\\theta_{n,\\vv}\\rightarrow\\theta_0~~~~\\mbox{ in probability as }\n \\vv\\rightarrow0 ~~\\mbox{ and }~~n\\rightarrow}\\def\\l{\\ell\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle.\n\\end{equation*}\n\n\\end{thm}\n\nFor $A:=(A_1,A_2,\\cdots,A_p)\\in\\R^p\\otimes\\R^{pd}$ with $A_{k}\\in\n\\R^p\\otimes\\R^d$, $k=1,\\cdots,p,$ and $B\\in\\R^d$, define $A\\circ\nB\\in\\R^p\\otimes\\R^p$ by\n\\begin{equation*}\nA\\circ B=(A_1B,A_2B,\\cdots,A_pB).\n\\end{equation*}\nFor any $\\theta\\in\\Theta$, set\n\\begin{equation}\\label{0z3}\nI(\\theta):=\\int_0^T(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*(X_t^0,\\mathscr{L}_{X_t^0},\\theta)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)(X_t^0,\\mathscr{L}_{X_t^0},\\theta)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t,\n\\end{equation}\n\\begin{equation}\\label{0z2}\n\\begin{split}\nK(\\theta):&=-2\\int_0^T(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\nb^*)(X_t^0,\\mathscr{L}_{X_t^0},\\theta)\\circ\\Big(\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\n\\Gamma(X_t^0,\\theta,\\theta_0)\\Big)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t,\n\\end{split}\n\\end{equation}\nwhere $(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)} b^*):=(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^*))$, and,\nfor any random variable $\\zeta\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$ with\n$\\mathscr{L}_\\zeta\\in\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$,\n\\begin{equation}\\label{0s0}\n\\Upsilon(\\zeta,\\theta_0)=(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*(\\zeta,\\mathscr{L}_{\\zeta},\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta,\\mathscr{L}_{\\zeta}).\n\\end{equation}\n\nAnother main result in this paper is presented as below, which\nreveals the asymptotic distribution of $\\hat\\theta_{n,\\vv}.$\n\n\n\\begin{thm}\\label{th2}\n Let the assumptions of Theorem \\ref{th1} hold and suppose\nfurther that $({\\bf C})$ holds and that $I(\\cdot)$ and $K(\\cdot)$\ndefined in \\eqref{0z3} and \\eqref{0z2}, respectively, are\ncontinuous. Then,\n\\begin{equation*}\n\\vv^{-1}(\\hat\\theta_{n,\\vv}-\\theta_0)\\rightarrow\nI^{-1}(\\theta_0)\\int_0^T\\Upsilon(X_t^0,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(t)~~~~\\mbox{ in\nprobability }\n\\end{equation*}\nas $\\vv\\rightarrow0$ and $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$, where $\\Upsilon(\\cdot)$\nis given in \\eqref{0s0}.\n\\end{thm}\n\n\nWith contrast to the existing literature, the innovations of this\npaper lie in:\n\\begin{itemize}\n\\item[(i)] The classical contrast function for LSE is based on EM\nalgorithm. Whereas, under the monotone condition, the EM scheme no\nlonger works. Hence in the present work we adopt a tamed EM method\nto establish the corresponding contrast function. The above is our\nfirst innovation.\n\n\\item[(ii)]For the classical setup, the discrete-time\nobservations at the gridpoints are sufficient to construct the\ncontrast function. Nevertheless, for our present model, the\ndiscrete-time observations are insufficient to establish the\ncontrast function since the SDEs involved are path-dependent. In\nthis paper, we overcome the difficulty mentioned by linear\ninterpolation w.r.t. the discrete-time observations. The above is\nour second innovation.\n\n\\item[(iii)] Our model is much more applicable, which allow the\ncoefficients to be distribution-dependent and weakly monotone. In\nparticular, the drift terms are allowed to be singular (e.g.,\nH\\\"older continuous). The above is our third innovation.\n\n\n\\end{itemize}\n\n\n\nNow, we provide a concrete example to demonstrate Theorems\n\\ref{th1} and \\ref{th2}.\n\\begin{exa}\\label{exa}\nFor any $\\vv\\in(0,1)$, consider the following scalar\npath-distribution dependent SDE\n\\begin{equation}\n\\begin{split}\n\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D X^\\vv(t)&=\\theta^{(1)}+\\theta^{(2)}\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\\Big(-\n(X^\\vv(t))^3+X^\\vv(t)+\\int_{-r_0}^0X^\\vv(t+\\theta)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\theta+\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\n \\zeta(\\theta) \\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\theta\\Big)\\mathscr{L}_{X^\\vv_t}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta)\\\\\n&\\quad+\\vv\\,\\Big(1+ \\int_{-r_0}^0X^\\vv(t+\\theta)|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\theta\\Big)\\,\n\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(t),~~~t\\ge0\n\\end{split}\n\\end{equation}\nwith the initial value $X_0^\\vv=\\xi\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$ which is Lipschitz,\nwhere, for some $c_10$ is a\ngeneric constant whose value may change from line to line.\n\n\n\\section{Proof of Theorem \\ref{th1}}\\label{sec2}\nTo complete the proof of Theorem \\ref{th1}, we provide some\ntechnical lemmas. The lemma below expounds that the path associated\nwith \\eqref{q3} is uniformly bounded in the $p$-th moment sense.\n\\begin{lem}\n Let $({\\bf A1})$ and $({\\bf A2})$ hold. Then, for any $p>0$ there\nis a constant $C_{p,T}>0$ such that\n\\begin{equation}\\label{r0}\n\\sup_{0\\le t\\le T}\\|X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^p\\le C_{p,T}(1+\\|\\xi\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^p),\n\\end{equation}\nand\n\\begin{equation}\\label{r6}\n\\sup_{0\\le t\\le T}\\E\\Big(\\sup_{-r_0\\le s\\le t}|Y^\\vv(s)|^p\\Big)\\le\nC_{p,T}(1+\\|\\xi\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^p).\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nWith the assumption ({\\bf A1}) in hand, the proof of \\eqref{r0} can\nbe achieved by the chain rule and the Gronwall inequality. We\nherein omit the details since it is\n standard. Now we turn to show the argument of\n\\eqref{r6}. By H\\\"older's inequality, it suffices to verify that\n\\eqref{r6} holds for any $p>4.$ By It\\^o's formula, we deduce that\n\\begin{equation*}\n\\begin{split}\n|Y^\\vv(t)|^p&=|Y^\\vv(0)|^p+\\int_0^t\\Big\\{p|Y^\\vv(s)|^{p-2}\\+\n\\ff{p}{2}|Y^\\vv(s)|^{p-2}\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}^*(\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^2\\\\\n&\\quad +\\ff{p(p-2)}{2}|Y^\\vv(s)|^{p-4}|\\sigma} \\def\\ess{\\text{\\rm{ess}}^*(\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})Y^\\vv(s)|^2\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+p\\int_0^t|Y^\\vv(s)|^{p-2}\\\\\\\n&\\le p\\int_0^t|Y^\\vv(s)|^{p-2}\\\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+p\\int_0^t|Y^\\vv(s)|^{p-2}\\\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+\\ff{p(p-1)}{2}\\int_0^t|Y^\\vv(s)|^{p-2}\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^2\n\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+p\\int_0^t|Y^\\vv(s)|^{p-2}\\\\\\\n&=:\\sum_{i=1}^4\\Pi_i(t),~~~~~~t\\in[0,T].\n\\end{split}\n\\end{equation*}\nWhence, for any $t\\ge0$ one has\n\\begin{equation}\\label{w6}\n\\begin{split}\n\\Upsilon(t):=\\E\\Big(\\sup_{-r_0\\le s\\le\nt}|Y^\\vv(s)|^p\\Big)\\le\\|\\xi\\|^p_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\sum_{i=1}^4\\E\\Big(\\sup_{0\\le\ns\\le t}\\Pi_i(s)\\Big).\n\\end{split}\n\\end{equation}\nIn the sequel, we are going to claim that\n\\begin{equation}\\label{w7}\n\\Upsilon(t) \\le 2\\|\\xi\\|^p_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+ c\\,t+c \\int_0^t \\Upsilon(s) \\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{equation}\nIf \\eqref{w7} was true, thus \\eqref{r6} follows directly from\nGronwall's inequality. So, it remains to verify that \\eqref{w7}\nholds true.\n\n\nLet $\\zeta_0(s)\\equiv {\\bf 0}\\in\\R^d$\nfor any $s\\in[-r_0,0].$ For $\\zeta\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$ and $\\mu\\in\\mathcal\n{P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$, we deduce from ({\\bf A1}) that\n\\begin{equation}\\label{r1}\n\\begin{split}\n\\<\\zeta(0),b(\\zeta,\\mu,\\theta)\\>&=\\<\\zeta(0)-\\zeta_0,b(\\zeta,\\mu,\\theta)-b(\\zeta_0,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0},\\theta)\\>\n+\\<\\zeta(0),b(\\zeta_0,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0},\\theta)\\>\\\\\n&\\le\\aa_1\\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\aa_2\\mathbb{W}_2(\\mu,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2+|\\zeta(0)|^2+|b(\\zeta_0,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0},\\theta)|^2\\\\\n&\\le c\\,(1+\\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mu,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2),\n\\end{split}\n\\end{equation}\nand from ({\\bf A2}) that\n\\begin{equation}\\label{r2}\n\\begin{split}\n\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta,\\mu)\\|^\n&\\le\n2\\bb_1\\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+2\\bb_2\\mathbb{W}_2(\\mu,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2+2\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta_0,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\|^2\\\\\n&\\le c\\,(1+\\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mu,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2).\n\\end{split}\n\\end{equation}\nAccording to \\eqref{w2}, we obtain that\n\\begin{equation}\\label{r5}\n\\begin{split}\n&\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\\\\n&=\\max_{k=0,\\cdots,M-1}\\sup_{-(k+1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le s\\le-k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv(s)|\\\\\n&=\\max_{k=0,\\cdots,M-1}\\sup_{-(k+1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le\ns\\le-k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\Big|\\ff{s+(k+1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}Y^\\vv(t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho-k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)-\\ff{s+k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}Y^\\vv(t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho-(k+1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\Big|\\\\\n&\\le 2\\sup_{-r_0\\le s\\le t}|Y^\\vv(s)|.\n\\end{split}\n\\end{equation}\nFurthermore, recall the Young inequality:\n\\begin{equation}\\label{r3}\na^\\aa b^{1-\\aa}\\le \\aa a+(1-\\aa)b,~~~~~a,b\\ge0,~~\\aa\\in[0,1],\n\\end{equation}\nand the fundamental fact that: for any $q>0$,\n\\begin{equation}\\label{w8}\n\\E|B(t)|^q\\le c\\, t^{q\/2}.\n\\end{equation}\nBy virtue of \\eqref{w2}, we notice that\n\\begin{equation}\\label{r8}\n\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv(0)=Y^\\vv(t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho).\n\\end{equation}\nThen, by exploiting \\eqref{r1}, \\eqref{r5} as well as \\eqref{r8}, it\nfollows from \\eqref{r3} and H\\\"older's inequality that\n\\begin{equation}\\label{w3}\n\\begin{split}\n&\\E\\Big(\\sup_{0\\le s\\le t}\\Pi_1(s)\\Big) \\\\&= p\\,\\E\\Big(\\sup_{0\\le\ns\\le t}\\int_0^s\\ff{|Y^\\vv(u)|^{p-2}}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa |b(\\bar\nY^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar\nY^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta)|}\\\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nu\\Big)\\\\\n&= p\\,\\E\\Big(\\sup_{0\\le s\\le\nt}\\int_0^s\\ff{|Y^\\vv(u)|^{p-2}}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa |b(\\bar\nY^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta)|}\\<\\bar\nY^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}(0),b(\\bar Y^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar\nY^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta)\\>\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nu\\Big)\\\\\n&\\le c\\,\\E\\Big(\\sup_{0\\le s\\le\nt}\\int_0^s\\ff{|Y^\\vv(u)|^{p-2}}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa| b(\\bar\nY^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar\nY^\\vv_{u_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta)|}\\Big\\{1+\\|\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le\nc\\int_0^t\\Big\\{1+\\E|Y^\\vv(s)|^p+\\E\\|\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^p\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\le c\\int_0^t\\{1+\\Upsilon(s)\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation}\nIt is straightforward to see that, for any $\\zeta\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$,\n$\\mu\\in\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$, and $\\theta\\in\\Theta$,\n\\begin{equation}\\label{r4}\n|b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta,\\mu,\\theta)|=\\ff{|b(\\zeta,\\mu,\\theta)|}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta,\\mu,\\theta)|}\\le\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{-\\aa}.\n\\end{equation}\nTaking \\eqref{r2} and \\eqref{r4} into consideration and making use\nof \\eqref{w8} and $\\aa\\in(0,1\/2]$, for any $q\\ge2$, we derive that\n\\begin{equation}\\label{r7}\n\\begin{split}\n\\E|Y^\\vv(t)-Y^\\vv(t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^q&\\le\nc\\,\\Big\\{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{q(1-\\aa)}+\\E\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^q\\E|B(t)-B(t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^q\\Big\\}\\\\\n&\\le\nc\\,\\Big\\{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{q(1-\\aa)}+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{q\/2}\\E\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^q\\Big\\}\\\\\n&\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{q\/2}\\Big\\{1+\n\\E\\|\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|^q+\\mathbb{W}_2(\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^q\\Big\\}\\\\\n&\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{q\/2}\\Big\\{1+ \\E\\Big(\\sup_{-r_0\\le s\\le\nt}|Y^\\vv(s)|^q\\Big)\\Big\\},\n\\end{split}\n\\end{equation}\nwhere in the last procedure we have used H\\\"older's inequality and\n\\eqref{r5}. Thus, taking advantage of \\eqref{r4} and \\eqref{r7} and\nemploying H\\\"older's inequality yields that\n\\begin{equation}\\label{w4}\n\\begin{split}\n\\E\\Big(\\sup_{0\\le s\\le t}|\\Pi_2(s)|\\Big)&\\le\np\\E\\int_0^t|Y^\\vv(s)|^{p-2}|Y^\\vv(s)-Y^\\vv(s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|\\cdot|b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta)|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le\np\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{-\\aa}\\int_0^t\\E(|Y^\\vv(s)|^{p-2}|Y^\\vv(s)-Y^\\vv(s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\le\np\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{-\\aa}\\int_0^t\\Big(\\E(|Y^\\vv(s)|^p)\\Big)^{\\ff{p-2}{p}}\\Big(\\E|Y^\\vv(s)-Y^\\vv(s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^{\\ff{p}{2}}\\Big)^{\\ff{2}{p}}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le\np\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\ff{1}{2}-\\aa}\\int_0^t\\Big(\\E(|Y^\\vv(s)|^p)\\Big)^{\\ff{p-2}{p}}\\Big\\{1+\n\\E\\Big(\\sup_{-r_0\\le s\\le\nt}|Y^\\vv(s)|^{\\ff{p}{2}}\\Big)\\Big\\}^{\\ff{2}{p}}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\le c \\int_0^t\\{1+\\Upsilon(s)\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s,\n\\end{split}\n\\end{equation}\nwhere in the last display we used\n $\\aa\\in(0,1\/2]$ and \\eqref{r3}. Next, we observe\nthat\n\\begin{equation}\\label{q1}\n\\begin{split}\n\\E\\Big(\\sup_{0\\le s\\le t}\\Pi_3(s)\\Big)&\\le\n\\ff{p(p-1)}{2}\\int_0^t\\E(|Y^\\vv(s)|^{p-2}\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^2) \\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation}\nUsing Burkhold-Davis-Gundy's (BDG's for short) inequality and\n\\eqref{r3}, we infer that\n\\begin{equation}\\label{q2}\n\\begin{split}\n\\E\\Big(\\sup_{0\\le s\\le t}\\Pi_4(s)\\Big)&\\le p\\,\\E\\Big(\\sup_{0\\le s\\le\nt}\\Big|\\int_0^s|Y^\\vv(u)|^{p-2}\\\\Big|\\Big)\\\\\n&\\le 4\\ss2\\,p\\,\\E\\Big(\\int_0^t|Y^\\vv(s)|^{2(p-2)}|\\sigma} \\def\\ess{\\text{\\rm{ess}}^*(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})Y^\\vv(s)|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{1\/2}\\\\\n&\\le 4\\ss2\\,p\\,\\E\\Big(\\sup_{0\\le s\\le\nt}|Y^\\vv(s)|^p\\int_0^t|Y^\\vv(s)|^{p-2}\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{1\/2}\\\\\n&\\le\\ff{1}{2}\\Upsilon(t)+16p^2\\int_0^t\\E(|Y^\\vv(s)|^{p-2}\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^2)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation}\nSubsequently, one gets from \\eqref{q1} and \\eqref{q2} that\n\\begin{equation}\\label{w5}\n\\begin{split}\n&\\E\\Big(\\sup_{0\\le s\\le t}\\Pi_3(s)\\Big)+\\E\\Big(\\sup_{0\\le s\\le\nt}\\Pi_4(s)\\Big)\\\\&\\le\n\\ff{1}{2}\\Upsilon(t)+c\\int_0^t\\E(|Y^\\vv(s)|^{p-2}\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^2)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le \\ff{1}{2}\\Upsilon(t)+ c\\int_0^t\\{\\E|Y^\\vv(s)|^p+\\E\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^p\\}\n\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\le \\ff{1}{2}\\Upsilon(t)+ c\\int_0^t\\Big\\{1+\\E|Y^\\vv(s)|^p+\\E\\|\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^p+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^p\\Big\\}\n\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\le \\ff{1}{2}\\Upsilon(t)+ c \\int_0^t\\{1+\\Upsilon(s)\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s,\n\\end{split}\n\\end{equation}\nwhere we have adopted \\eqref{r3} in the second inequality, used\n\\eqref{r2} in the last two step, and utilized H\\\"older's inequality,\nin addition to \\eqref{r5}, in the last procedure. Substituting\n\\eqref{w3}, \\eqref{w4}, and \\eqref{w5} into \\eqref{w6} gives that\n\\begin{equation*}\n\\Upsilon(t) \\le \\|\\xi\\|^p_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+ \\ff{1}{2}\\Upsilon(t)+ c\n\\int_0^t\\{1+\\Upsilon(s)\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{equation*}\nAs a consequence, \\eqref{w7} is now available.\n\\end{proof}\n\n\nThe following lemma shows that the linear interpolation $\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}$ approaches $X_t^0$ in the mean square sense as\n $\\vv$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho$ go to zero.\n\\begin{lem}\\label{le1}\n Assume $({\\bf A1}), ({\\bf A2}), ({\\bf B1})$ and $({\\bf B4})$. Then,\nfor any $\\bb\\in(0,1)$, there exists $c_\\bb>0$\n\n\\begin{equation}\\label{a9}\n \\sup_{0\\le t\\le T}\\E\\|\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\le\n c_\\bb(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+\\vv^2+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa}),\n\\end{equation}\nwhere $\\aa\\in(0,1\/2]$ is introduced in \\eqref{e4}.\n\\end{lem}\n\n\\begin{proof}\nFor any $\\bb\\in(0,1)$ and $t\\in[0,T]$, by H\\\"older's inequality and\n$Y_0^\\vv=X_0^0=\\xi$, we find that\n\\begin{equation}\\label{a7}\n\\begin{split}\n\\E\\|\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2&\\le3\\,\\E\\|Y_t^\\vv-\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+3\\,\\E\\|Y^\\vv_t-X_t^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+3\\,\\E\\|X^\\vv_t-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\\\\n&\\le3\\,\\E\\Big(\\sup_{-r_0\\le v\\le 0}|Y^\\vv(t+v)-\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}(v)|^2\\Big)+3\\,\\E\\Big(\\sup_{0\\le s\\le\nt}|Y^\\vv(s)-X^\\vv(s)|^2\\Big)\\\\\n&\\quad+3\\,\\E\\Big(\\sup_{0\\le s\\le t}|X^\\vv(s)-X^0(s)|^2\\Big)\\\\\n&\\le3\\,\nM^{1-\\bb}\\max_{k=0,\\cdots,M-1}\\,\\Big(\\E\\Big(\\sup_{-(k+1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le\nv\\le-k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}| Y^\\vv(t+v)-\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}(v)|^{\\ff{2}{1-\\bb}}\\Big)\\Big)^{1-\\bb}\\\\\n&\\quad+3\\,\\E\\Big(\\sup_{0\\le s\\le t}|Y^\\vv(s)-X^\\vv(s)|^2\\Big)\n+3\\,\\E\\Big(\\sup_{0\\le s\\le\nt}|X^\\vv(s)-X^0(s)|^2\\Big)\\\\\n&=:\\LL_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\LL_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho),\n\\end{split}\n\\end{equation}\nwhere $M>0$ such that $M\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho=r_0.$ Hereinafter, we intend to estimate\n$\\LL_i(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)$, $i=1,2,3,$ respectively. In the first place, we\nshall show that\n\\begin{equation}\\label{e3}\n\\LL_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb,~~~~t\\in[0,T].\n\\end{equation}\nFor $t\\in[0,T)$, there is an integer $k_0\\ge0$ such that\n$t\\in[k_0\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho,(k_0+1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho).$ From \\eqref{w2}, it follows that\n\\begin{equation}\\label{a00}\n\\begin{split}\n&\\LL_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\\\&\\le c\\,\nM^{1-\\bb}\\max_{k=0,\\cdots,M-1}\\,\\Big(\\E\\Big(\\sup_{(k_0-k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le\ns\\le(k_0+1-k)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|\nY^\\vv(s)-Y^\\vv((k_0-k)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^{\\ff{2}{1-\\bb}}\\Big)\\Big)^{1-\\bb}\\\\\n&\\quad+\n c\\,\nM^{1-\\bb}\\max_{k=0,\\cdots,M-1}\\,\\Big(\\E\\Big(\\sup_{(k_0-k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le\ns\\le(k_0+1-k)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|\nY^\\vv(s)-Y^\\vv((k_0-k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^{\\ff{2}{1-\\bb}}\\Big)\\Big)^{1-\\bb}\\\\\n&\\le\n c\\,\nM^{1-\\bb}\\max_{k=0,\\cdots,M-1}\\,\\Big(\\E\\Big(\\sup_{(k_0-k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le\ns\\le(k_0+1-k)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|\nY^\\vv(s)-Y^\\vv((k_0-k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^{\\ff{2}{1-\\bb}}\\Big)\\Big)^{1-\\bb}\\\\\n&\\quad+c\\, M^{1-\\bb}\\max_{k=0,\\cdots,M-1}\\,\\Big(\\E|\nY^\\vv((k_0-k)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)-Y^\\vv((k_0-k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^{\\ff{2}{1-\\bb}}\\Big)^{1-\\bb}.\n\\end{split}\n\\end{equation}\nIn case of $k\\ge k_0+1$, by virtue of ({\\bf B4}), one has\n\\begin{equation*}\n\\LL_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le c\\,M^{1-\\bb}\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^2\\le c\\,r_0^{1-\\bb}\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb.\n\\end{equation*}\nIn terms of ({\\bf B1}), for any $\\zeta\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$ and $\\mu\\in\\mathcal\n{P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$,\n\\begin{equation}\\label{q4}\n\\begin{split}\n|b(\\zeta,\\mu,\\theta_0)|&\\le|b(\\zeta,\\mu,\\theta_0)-b(\\zeta_0,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0},\\theta_0)|+|b(\\zeta_0,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0},\\theta_0)|\\\\\n&\\le\nL_1\\Big\\{(1+\\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_1})\\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mu,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big\\}+|b(\\zeta_0,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0},\\theta_0)|.\n\\end{split}\n\\end{equation}\nLet $k'\\ge0$ be an arbitrary integer. For any\n$t\\in[k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho,(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho]$, note from BDG's inequality followed by\nH\\\"older's inequality that\n\\begin{equation*}\n\\begin{split}\n&\\E\\Big(\\sup_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le t\\le\n(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|Y^\\vv(t)-Y^\\vv(k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^{\\ff{2}{1-\\bb}}\\Big)\\\\&\\le\nc\\,\\E\\Big(\\int_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^{(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta_0)|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{\\ff{2}{1-\\bb}}+c\\,\\E\\Big(\\sup_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le t\\le\n(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\Big|\\int_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^t\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nB(s)\\Big|^{\\ff{2}{1-\\bb}}\\Big)\\\\\n&\\le c\\,\\E\\Big(\\int_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^{(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta_0)|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{\\ff{2}{1-\\bb}}+c\\,\\E\\Big(\\int_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^{(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{\\ff{1}{1-\\bb}}\\\\\n&\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\ff{\\bb}{1-\\bb}}\\int_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^{(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\Big\\{\\E|b(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\theta_0)|^{\\ff{2}{1-\\bb}}+ \\E\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^{\\ff{2}{1-\\bb}}\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s,\n\\end{split}\n\\end{equation*}\nwhere in the last display we have used the fact that\n\\begin{equation}\\label{t7}\n|b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta,\\mu,\\theta_0)|\\le|b(\\zeta,\\mu,\\theta_0)|,~~~~\\zeta\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r,~~~\\mu\\in\\mathcal\n{P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r).\n\\end{equation}\nSubsequently, taking \\eqref{r6}, \\eqref{r2} and \\eqref{q4} into\naccount and making use of H\\\"older's inequality yields that\n\\begin{equation}\\label{d1}\n\\begin{split}\n&\\E\\Big(\\sup_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le t\\le\n(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|Y^\\vv(t)-Y^\\vv(k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^{\\ff{2}{1-\\bb}}\\Big)\\\\\n&\\le\nc\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\ff{\\bb}{1-\\bb}}\\int_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^{(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\Big\\{1+\\E\\|\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{\\ff{2(1+q_1)}{1-\\bb}}+\n\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^{\\ff{2}{1-\\bb}}\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le\nc\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\ff{\\bb}{1-\\bb}}\\int_{k'\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^{(k'+2)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\Big\\{1+\\E\\|\\bar\nY^\\vv_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{\\ff{2(1+q_1)}{1-\\bb}}\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\ff{1}{1-\\bb}}.\n\\end{split}\n\\end{equation}\nHence, it follows from \\eqref{a00} and \\eqref{d1} with\n$k^\\prime=k_0-k-1$ that\n\\begin{equation*}\n\\begin{split}\n\\LL_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le c\\, M^{1-\\bb}\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb\n\\end{split}\n\\end{equation*}\nprovided that $k\\le k_0-1$. Whenever $k=k_0$, we deduce from\n\\eqref{a00}, \\eqref{d1} with $k'=0$ as well as ({\\bf B4}) that\n\\begin{equation*}\n\\begin{split}\n\\LL_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)&\\le c\\, M^{1-\\bb}\\Big(\\E\\Big(\\sup_{0\\le s\\le\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}|\nY^\\vv(s)-Y^\\vv(0)|^{\\ff{2}{1-\\bb}}\\Big)\\Big)^{1-\\bb}\\\\\n&\\quad+ c\\, M^{1-\\bb}\\Big(\\E\\Big(\\sup_{-\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\le s\\le0}|\nY^\\vv(s)-Y^\\vv(-\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^{\\ff{2}{1-\\bb}}\\Big)\\Big)^{1-\\bb}\\\\\n&\\quad+c\\, M^{1-\\bb}|\nY^\\vv(0)-Y^\\vv(-\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|^2\\\\\n&\\le c\\, M^{1-\\bb}\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\\\\n&\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb.\n\\end{split}\n\\end{equation*}\nNext, we are going to claim that\n\\begin{equation}\\label{a8}\n\\LL_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho) \\le c\\,\\vv^2,~~~~t\\in[0,T].\n\\end{equation}\nFollowing the argument to derive \\eqref{r6}, we deduce that, for\nsome constant $C_{p,T}>0,$\n\\begin{equation}\\label{0r6}\n\\sup_{0\\le t\\le T}\\E\\|X^\\vv_t\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^p\\le C_{p,T}(1+\\|\\xi\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^p).\n\\end{equation}\nBy the It\\^o formula and $X^\\vv_0=X_0^0=\\xi$, we observe that\n\\begin{equation*}\n\\begin{split}\n&|X^\\vv(t)-X^0(t)|^2\\\\\n&=\\int_0^t\\{2\\+\\vv^2\\|\n\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_s^\\vv,\\mathscr{L}_{X_s^\\vv})\\|^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\quad+2\\,\\vv\\int_0^t\\.\n\\end{split}\n\\end{equation*}\nThus, by using BDG's inequality and \\eqref{r3} and noting that\n$X^\\vv_0=X_0^0=\\xi$, we infer from ({\\bf A1}) and \\eqref{r2} that\n\\begin{equation*}\n\\begin{split}\n\\LL_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)&\\le2\\int_0^t\\{\\aa_1\\E\\|X_s^\\vv-X_s^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\aa_2\\mathbb{W}_2(\\mathscr{L}_{X_s^\\vv},\\mathscr{L}_{X_s^0})^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\quad+c\\,\\vv^2\\int_0^t\\{1+\\E\\|X_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mathscr{L}_{X_s^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\quad+8\\ss2\\,\\vv\\E\\Big(\\int_0^t|\\sigma} \\def\\ess{\\text{\\rm{ess}}^*(X_s^\\vv,\\mathscr{L}_{X_s^\\vv})(X^\\vv(s)-X^0(s))|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{1\/2}\\\\\n&\\le c\\int_0^t\\LL_3(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s+c\\,\\vv^2\\int_0^t\\{1+\\E\\|X_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+8\\ss2\\,\\vv\\E\\Big(\\sup_{0\\le s\\le\nt}|X^\\vv(s)-X^0(s)|^2\\int_0^t\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_s^\\vv,\\mathscr{L}_{X_s^\\vv})\\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{1\/2}\\\\\n&\\le\\ff{1}{2}\\LL_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+c\\int_0^t\\LL_3(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns+c\\,\\vv^2\\int_0^t\\{1+\\E\\|X_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation*}\nSo, one has\n\\begin{equation*}\n\\LL_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho) \\le c\\int_0^t\\LL_3(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns+c\\,\\vv^2\\int_0^t\\{1+\\E\\|X_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{equation*}\nThus, \\eqref{a8} follows from \\eqref{0r6} and Gronwall's inequality.\nFinally, we intend to verify that\n\\begin{equation}\\label{s1}\n\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le c\\,(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa}),~~~~t\\in[0,T].\n\\end{equation}\nAlso, by It\\^o's formula, we derive from $X_0^\\vv=Y_0^\\vv=\\xi$ that\n\\begin{align*}\n|X^\\vv(t)-Y^\\vv(t)|^2&=2\\int_0^t\\\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+2\\int_0^t\\\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+2\\int_0^t\\\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+\\vv^2\\int_0^t\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_s^\\vv,\\mathscr{L}_{X_s^\\vv})-\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})\\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\quad+2\\,\\vv\\int_0^t\\\\\\\n&=:\\Xi_1(t)+\\Xi_2(t)+\\Xi_3(t)+\\Xi_4(t)+\\Xi_5(t).\n\\end{align*}\nIn view of ({\\bf A1}), we deduce that\n\\begin{equation}\\label{a22}\n\\begin{split}\n\\E\\Big(\\sup_{0\\le s\\le\nt}\\Xi_1(s)\\Big)&\\le2\\int_0^t\\{\\aa_1\\E\\|X_s^\\vv-Y_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\aa_2\\mathbb{W}_2(\\mathscr{L}_{X_s^\\vv},\\mathscr{L}_{Y_s^\\vv})^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le c\\,\\int_0^t\\E\\|X_s^\\vv-Y_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\le c\\int_0^t\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation}\nCarrying out a similar argument to derive \\eqref{e3}, for any\n$\\kk>2$, we have\n\\begin{equation}\\label{w0}\n\\sup_{0\\le t\\le T}\\E\\|Y^\\vv_t-\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^\\kk\\le\nc\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\ff{\\kk}{2}-1}.\n\\end{equation}\nTaking ({\\bf A1}), \\eqref{r6} and \\eqref{w0} into consideration\nand applying H\\\"older's inequality that\n\\begin{equation}\\label{a2}\n\\begin{split}\n&\\E\\Big(\\sup_{0\\le s\\le\nt}|\\Xi_2(s)|\\Big)\\\\&\\le\\int_0^t\\{\\E|X^\\vv(s)-Y^\\vv(s)|^2+\n\\E|b(Y_s^\\vv,\\mathscr{L}_{Y_s^\\vv},\\theta_0)-b(\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta_0)|^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le c\\int_0^t\\E|X^\\vv(s)-Y^\\vv(s)|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\quad+c\\int_0^t\\E\\{(1+\\| Y_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2q_1}+\\|\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2q_1})\\|Y^\\vv_s-\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mathscr{L}_{Y_s^\\vv},\\mathscr{L}_{\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})^2\\} \\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le c\\int_0^t\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s+c\\int_0^t\\Big(\\E\\|Y^\\vv_s-\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{\\ff{2}{1-\\bb}}\\Big)^{1-\\bb}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+c\\int_0^t\\Big(\\E\\Big(1+\\| Y_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2q_1}+\\|\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2q_1}\\Big)^{\\ff{1}{\\bb}}\\Big)^{\\bb}\\Big(\\E\\|Y^\\vv_s-\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{\\ff{2}{1-\\bb}}\\Big)^{1-\\bb}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+c\\int_0^t\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation}\nAccording to \\eqref{e4} and in view of \\eqref{r6} and \\eqref{q4}, it\nfollows from H\\\"older's inequality that\n\\begin{equation}\\label{a3}\n\\begin{split}\n&\\E\\Big(\\sup_{0\\le s\\le\nt}|\\Xi_3(s)|\\Big)\\\\&\\le2\\int_0^t\\E\\{|X^\\vv(s)-Y^\\vv(s)|\\cdot\n|b(\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta_0)-b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta_0)|\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\le c\\int_0^t\\E\\Big\\{|X^\\vv(s)-Y^\\vv(s)|^2+\n \\ff{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa}|b(\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta_0)|^4}{(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|\n b(\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta_0)|)^2}\n \\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n &\\le c\\int_0^t\\Big\\{\\E|X^\\vv(s)-Y^\\vv(s)|^2+\n\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa}\\E|b(\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta_0)|^4\n \\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n &\\le c\\int_0^t\\Big\\{\\E|X^\\vv(s)-Y^\\vv(s)|^2+\n\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa}\\{1+\\E\\|\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+q_1)}+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^4\\}\n \\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n &\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa}+c\\int_0^t\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation}\n Next, owing to $\\vv\\in(0,1)$, $({\\bf A2})$, and\n\\eqref{e3}, one gets that\n\\begin{equation}\\label{a1}\n\\begin{split}\n\\E\\Big(\\sup_{0\\le s\\le t}\\Xi_4(s)\\Big)&\\le\nc\\int_0^t\\{\\E\\|X_s^\\vv-\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mathscr{L}_{X_s^\\vv},\\mathscr{L}_{\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le c\\int_0^t\\{\\E\\|X_s^\\vv-Y_s^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\E\\|Y_s^\\vv-\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le c\\int_0^t\\{\\LL_1(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+c\\int_0^t\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation}\nNext, for $\\vv\\in(0,1)$, BDG's inequality and Young's inequality\n\\eqref{r3}, besides \\eqref{a1}, give that\n\\begin{equation}\\label{a5}\n\\begin{split}\n&\\E\\Big(\\sup_{0\\le s\\le\nt}|\\Xi_5(s)|\\Big)\\\\&\\le8\\ss2\\,\\E\\Big(\\int_0^t\n|(\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_s^\\vv,\\mathscr{L}_{X_s^\\vv})-\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv}))^*(X^\\vv(s)-Y^\\vv(s))|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{1\/2}\\\\\n&\\le8\\ss2\\,\\E\\Big(\\sup_{0\\le\\le\nt}|X^\\vv(s)-Y^\\vv(s)|^2\\int_0^t\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_s^\\vv,\\mathscr{L}_{X_s^\\vv})-\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})\\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\Big)^{1\/2}\\\\\n&\\le\n\\ff{1}{2}\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+c\\int_0^t\\E\\|\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_s^\\vv,\\mathscr{L}_{X_s^\\vv})-\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})\\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\ns\\\\\n&\\le\n\\ff{1}{2}\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+c\\int_0^t\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{split}\n\\end{equation}\nThus, \\eqref{a22}, \\eqref{a2}-\\eqref{a5} yield that\n\\begin{equation*}\n\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le\\ff{1}{2}\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\nc\\,(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa})+c\\int_0^t\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{equation*}\nNamely,\n\\begin{equation*}\n\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le\nc\\,(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa})+c\\int_0^t\\LL_2(s,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s.\n\\end{equation*}\nAs a result, we obtain from Gronwall's inequality that\n\\begin{equation}\\label{a6}\n\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le c\\,(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{2\\aa}).\n\\end{equation}\nInserting \\eqref{e3}, \\eqref{a8}, and \\eqref{a6} back into\n\\eqref{a7} leads to the desired assertion \\eqref{a9}.\n\\end{proof}\n\n\n\\begin{rem}\n{\\rm The convergence rate of EM scheme for path-independent SDEs\nunder the global Lipschitz condition is one half. Taking $\\aa=1\/2$\nin \\eqref{s1}, we conclude that the convergence rate of the tamed\nEM scheme constructed in \\eqref{q3} is close sufficiently to one\nhalf. This demonstrate the distinct features between path-dependent\nSDEs and path-independent SDEs. }\n\\end{rem}\n\n\nThe lemma below plays a crucial role in revealing the asymptotic\nbehavior of the LSE of the unknown parameter $\\theta\\in\\Theta$.\n\n\n\n\n\n\n\n\n\\begin{lem}\\label{lem}\n Let $({\\bf A1})-({\\bf A2})$ and $({\\bf B1})-({\\bf B4})$ hold. Then,\n\\begin{equation}\\label{t4}\n\\begin{split}\n&\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\sum_{k=1}^n(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\\\\n&\\rightarrow\\Xi(\\theta):=\\int_0^T\\Gamma(X_t^0,\\theta,\\theta_0)^*\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\Gamma(X_s^0,\\theta,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt\n\\end{split}\n\\end{equation}\nin $L^1$ as $\\vv\\rightarrow0$ and $\\delta\\rightarrow0 $ $($i.e.,\n$n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$$)$. Moreover,\n\\begin{equation}\\label{q6}\n \\sum_{k=1}^n(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta_0) \\rightarrow0\n\\end{equation}\nin $L^2$ as $\\vv\\rightarrow0$.\n\\end{lem}\n\n\n\\begin{proof}\nObserve that\n\\begin{equation*}\n\\begin{split}\n&\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\sum_{k=1}^n(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)-\n\\int_0^T\\Gamma^*(X_t^0,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\Gamma(X_t^0,\\theta,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt\\\\\n&=\\int_0^T\\Big\\{(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\n-\\Gamma^*(X_t^0,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\Gamma(X_t^0,\\theta,\\theta_0)\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt\\\\\n&=\\int_0^T \\Big(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)-\\Gamma(X_t^0,\\theta,\\theta_0)\\Big)^*\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\quad+\\int_0^T\\Gamma(X_t^0,\\theta,\\theta_0)^*\n \\Big(\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)-\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\Big)\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0) \\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt\\\\\n&\\quad+\\int_0^T\\Gamma(X_t^0,\\theta,\\theta_0)^*\n\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\Big(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)-\\Gamma(X_t^0,\\theta,\\theta_0)\\Big) \\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt\\\\\n&=:J_1(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+J_2(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+J_3(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho).\n\\end{split}\n\\end{equation*}\nFrom ({\\bf B1}) and \\eqref{q4}, a direct calculation shows that,\nfor any random variables $\\zeta_1,\\zeta\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$ with\n$\\mathscr{L}_{\\zeta_1},\\mathscr{L}_{\\zeta_2}\\in\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$,\n\\begin{equation}\\label{t1}\n\\begin{split}\n&|\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta_1,\\theta,\\theta_0)-\\Gamma(\\zeta_2,\\theta,\\theta_0)|\\\\\n&=|b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)-b(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta_0)+\nb(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)-b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)\n|\\\\\n&\\le|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)-b(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta_0)|\n+|b(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)-b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|\\\\\n&\\quad+|b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)-b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)|\n+|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)-b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|\\\\\n&=|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)-b(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta_0)|\n+|b(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)-b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|\\\\\n&\\quad+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa\\Big|\\ff{|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)|}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)|}b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)\\Big|\n+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa\\Big|\\ff{|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|}\nb(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)\\Big|\\\\\n&\\le|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)-b(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta_0)|\n+|b(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)-b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|\\\\\n&\\quad+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\aa}\\{|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta_0)|^2+|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|^2\\}\\\\\n&\\le\nc\\Big\\{(1+\\|\\zeta_1\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_1}+\\|\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_1})\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mathscr{L}_{\\zeta_1},\\mathscr{L}_{\\zeta_2})\\Big\\}\\\\\n&\\quad+c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\aa}\\Big\\{1+\\|\\zeta_1\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q_1)}+\\mathbb{W}_2(\\mathscr{L}_{\\zeta_1},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2\\Big\\}.\n\\end{split}\n\\end{equation}\nNext, for a random variable $\\zeta\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$ with\n$\\mathscr{L}_\\zeta\\in\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$, by \\eqref{q4} and\n\\eqref{t7}, it follows that\n\\begin{equation}\\label{t3}\n\\begin{split}\n|\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta,\\theta,\\theta_0)|+|\\Gamma(\\zeta,\\theta,\\theta_0)\n\\le c\\,\\Big\\{1+\n \\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_1} +\n \\mathbb{W}_2(\\mathscr{L}_\\zeta,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big\\}\n \\end{split}\n\\end{equation}\nand, due to ({\\bf B3}), that\n\\begin{equation}\\label{t2}\n\\|\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)\\|\\le\\|\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)-\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(0)\\|+\\|\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(0)\\|\n\\le c\\,\\Big\\{1+\n \\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_3} +\n \\mathbb{W}_2(\\mathscr{L}_\\zeta,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big\\}.\n\\end{equation}\nConsequently, combining \\eqref{t1} with \\eqref{t3} and\\eqref{t2},\nfor $q:=q_1\\vee q_3$, we deduce from \\eqref{r0} that\n\\begin{align*}\n&|J_1(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|+|J_3(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|\\\\&\\le c\\int_0^T\\Big\\{(1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_1}+\\|X_s^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_1})\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mathscr{L}_{\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\mathscr{L}_{X_t^0})\\\\\n&\\quad+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\aa}\\Big(1+\\|\\bar\nY_{s_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q_1)}+\\mathbb{W}_2(\\mathscr{L}_{\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2\\Big)\\Big\\}\\\\\n&\\quad\\times\\Big\\{1 +\\|X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_1}+\n \\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_1} + \\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_0)\\Big\\}\\\\\n&\\quad\\times\\Big\\{1 +\\|X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_3}+\n \\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_3} + \\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\le c\\int_0^T\\Big\\{(1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^q)\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\ss{\\E\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2}\\Big\\}\\\\\n&\\quad\\times\\Big\\{1+\n \\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+ q)} + \\E\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n&\\quad+c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\aa}\\int_0^T\\Big\\{1+\n \\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+ q)} + \\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^4\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t.\n\\end{align*}\nThis, by exploiting \\eqref{r6} and \\eqref{a7} and using H\\\"older's\ninequality, gives that\n\\begin{equation}\\label{t5}\n\\begin{split}\n&\\E|J_1(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|+\\E|J_3(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|\\\\&\\le c\\,\\int_0^T\\ss{\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2}\\Big\\{1 +\\E\n \\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{8(1+ q)}\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\quad+c\\,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\aa}\\int_0^T\\Big\\{1+\\E\n \\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+ q)} \\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\rightarrow0\n\\end{split}\n\\end{equation}\n as $ \\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0$.\nNext, making use of ({\\bf B3}) and \\eqref{t3}, we derive that\n\\begin{equation*}\n\\begin{split}\n |J_2(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|&\\le c\\int_0^T(1+\n \\|X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_1})\\Big(1+\n \\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_1} + \\ss{\\E\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2}\\Big)\\\\\n&\\quad\\times\\Big((1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_3}+\\|X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_3})\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\ss{\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2}\\Big)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t.\n\\end{split}\n\\end{equation*}\nAgain, using \\eqref{r0}, \\eqref{r6} and \\eqref{a9} and utilizing\nH\\\"older's inequality gives that\n\\begin{equation}\\label{t6}\n\\begin{split}\n\\E|J_2(\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)|&\\le c\\,\\int_0^T\\ss{\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2}\\Big\\{1 +\\E\n \\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+ q)}\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\rightarrow0\n\\end{split}\n\\end{equation}\n as $ \\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0$.\nHence, \\eqref{t4} follows immediately from \\eqref{t5} and\n\\eqref{t6}.\n\n\nIn the sequel, we are going to show that \\eqref{q6} holds. In terms\nof \\eqref{q3}, we obtain that\n\\begin{equation}\\label{q5}\n\\begin{split}\n&\\sum_{k=1}^n(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta_0)\\\\\n&=\\vv\\sum_{k=1}^n(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\n\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y^\\vv_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})(B(k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)-B((k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho))\\\\\n&=\\vv\\int_0^T(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv) \\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(t).\n\\end{split}\n\\end{equation}\nBy the It\\^o isometry and the H\\\"older inequality, we derive from\n\\eqref{r2}, \\eqref{t3}, and \\eqref{t2} that\n\\begin{equation*}\n\\begin{split}\n&\\E\\Big|\\int_0^T(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv) \\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nB(t)\\Big|^2\\\\\n&=\\int_0^T\\E|(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv) \\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\n t\\\\\n &\\le\\int_0^T\\E\\{|\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)|^2\\cdot\\|\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\|^2\\cdot\\| \\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho},\\mathscr{L}_{\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}})\\|^2\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\le c\\,\\int_0^T\\E\\Big\\{\\Big(1+\\|\\bar\nY^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2\\Big)\\\\\n&\\quad\\times\\Big(1+\n \\|\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q_3)} +\n \\mathbb{W}_2(\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2\\Big)\\\\\n&\\quad\\times\\Big(1+\n \\|\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q_1)} +\n \\mathbb{W}_2(\\mathscr{L}_{\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2\\Big)\\Big\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\le c\\,\\int_0^T\\{1+\\E\\|\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{8(1+q)}\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t.\n\\end{split}\n\\end{equation*}\nThis, together with \\eqref{r6}, leads to\n\\begin{equation*}\n\\begin{split}\n&\\E\\Big|\\sum_{k=1}^n(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta_0)\\Big|^2\\\\\n&\\le c\\,\\vv^2\\int_0^T\\{1+\\E\\|\\bar Y^\\vv_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{8(1+q)}\\}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\le c\\,\\vv^2.\n\\end{split}\n\\end{equation*}\nAs a consequence, we obtain \\eqref{q6} immediately.\n\\end{proof}\n\n\n\nSo far, with Lemma \\ref{lem} in hand, we are in the position to\ncomplete the\n\\begin{proof}[ Proof of Theorem \\ref{th1}]\nA direction calculation shows that\n\\begin{equation}\\label{h1}\n\\begin{split}\n&\\Phi_{n,\\vv}(\\theta)\\\\\n&=\\delta^{-1}\\sum_{k=1}^n\\Big\\{P_k^*(\\theta)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta)-P_k^*(\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta_0)\\Big\\}\\\\\n&=\\delta^{-1}\\sum_{k=1}^n\\Big\\{\\Big(P_k(\\theta_0)+(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\Big)^*\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv) \\Big(P_k(\\theta_0)\n+\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y_{t_{k-1}}^\\vv,\\theta,\\theta_0)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\Big)\\\\\n&\\quad -P_k^*(\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta_0)\\Big\\}\\\\\n&=2\\sum_{k=1}^n(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta_0)\\\\\n&\\quad+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\sum_{k=1}^n(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0).\n\\end{split}\n\\end{equation}\nBy virtue of Lemma \\ref{lem}, we therefore infer from Chebyshev's\ninequality that\n\\begin{equation*}\n\\sup_{\\theta\\in\\Theta}|-\\Phi_{n,\\vv}(\\theta)-(-\\Xi(\\theta))|\\rightarrow0~~~~\\mbox{\nin probability.}\n\\end{equation*}\n Next, for any\n$\\kk>0,$ due to $\\Xi(\\cdot)>0$,\n\\begin{equation*}\n \\sup_{|\\theta-\\theta_0|\\ge\\kk}(-\\Xi(\\theta))<-\\Xi(\\theta_0)=0.\n\\end{equation*}\nFurthermore, one has $-\\Phi_{n,\\vv}(\n\\hat\\theta_{n,\\vv})\\ge-\\Phi_{n,\\vv}(\\theta_0)=0$. Consequently, we\ndeduce from \\cite[Theorem 5.9]{V98} with\n$M_n(\\cdot)=-\\Phi_{n,\\vv}(\\cdot)$ and $M(\\cdot)=-\\Xi(\\cdot)$ therein\nthat $\\hat\\theta_{n,\\vv}\\rightarrow\\theta_0$ in probability as\n$\\vv\\rightarrow0$ and $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$. We henceforth complete the\nproof.\n\\end{proof}\n\n\n\n\n\\section{Proof of Theorem \\ref{th2}}\\label{sec3}\nBefore we start to finish the argument of Theorem \\ref{th2}, we also\nneed to prepare some auxiliary lemmas below. For any random\nvariable $\\zeta\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$ with $\\mathscr{L}_\\zeta\\in\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$,\nset\n\\begin{equation*}\n\\Upsilon^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\zeta,\\theta):=(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*\n(\\zeta,\\mathscr{L}_\\zeta,\\theta)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta,\\mathscr{L}_{\\zeta}).\n\\end{equation*}\n\n\n\n\n\n\\begin{lem}\\label{le2}\n Let $({\\bf A1})-({\\bf A2})$ and $({\\bf B1})-({\\bf B4})$ hold. Then,\n\\begin{equation}\\label{v1}\n\\vv^{-1}(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Phi_{n,\\vv})(\\theta)\\rightarrow-2\\int_0^T\\Upsilon(X_t^0,\\theta)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nB(t)~~~~\\mbox{ in probability }\n\\end{equation}\nwhenever $\\vv\\rightarrow0$ and $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$, where\n$\\Upsilon(\\cdot,\\cdot)$ is introduced in \\eqref{0s0}.\n\\end{lem}\n\n\\begin{proof}\nBy the chain rule, one infers from \\eqref{q3} and \\eqref{h1} that\n\\begin{equation}\\label{c1}\n\\begin{split}\n&\\vv^{-1}(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Phi_{n,\\vv})(\\theta)\\\\%&=2\\vv^{-1}\\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta_0)\\\\\n&=2\\,\\vv^{-1}\\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\n\\Big\\{P_k(\\theta_0)+\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\Big\\}\\\\\n&=2\\,\\vv^{-1}\\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta)\\\\\n&=-2\\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})\\\\\n&\\quad\\times(B(k\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)-B((k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho))\\\\\n&=-2\\int_0^T\\Upsilon^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(t),\n\\end{split}\n\\end{equation}\nwhere in the last two display we used the fact that\n\\begin{equation}\\label{c2}\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)=-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}) (\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta).\n\\end{equation}\nTo achieve \\eqref{v1}, in terms of \\cite[Theorem 2.6, P.63]{F98}, it\nis sufficient to claim that\n\\begin{equation}\\label{v2}\n\\int_0^T\\|\\Upsilon^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta)-\\Upsilon(X_t^0,\\theta)\\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nt\\rightarrow0~~~~\\mbox{ in probability }\n\\end{equation}\nas $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0.$ Observe that\n\\begin{equation*}\n\\begin{split}\n&\\Upsilon^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta)-\\Upsilon(X_t^0,\\theta)\\\\\n&=(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^* (\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*\n(X_t^0,\\mathscr{L}_{X_t^0},\\theta)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0,\\mathscr{L}_{X_t^0})\\\\\n&=\\{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^* (\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*\n(X_t^0,\\mathscr{L}_{X_t^0},\\theta)\\}\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})\\\\\n&\\quad+(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*\n(X_t^0,\\mathscr{L}_{X_t^0},\\theta)\\{\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)-\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\}\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\n\\mathscr{L}_{\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})\\\\\n&\\quad+(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^* (X_t^0,\\mathscr{L}_{X_t^0},\\theta)\n\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0)\\{\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv})-\\sigma} \\def\\ess{\\text{\\rm{ess}}(X_t^0,\\mathscr{L}_{X_t^0})\\}\\\\\n&=:\\Sigma_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\Sigma_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\Sigma_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho).\n\\end{split}\n\\end{equation*}\n\n\n\n\n\n\n\n\n\n\n\nBy a straightforward calculation, for any random variable\n$\\zeta\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$ with $\\mathscr{L}_{\\zeta}\\in\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$, one\nhas\n\\begin{equation}\\label{v3}\n\\begin{split}\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})(\\zeta,\\mathscr{L}_\\zeta,\\theta)&=\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Big(\\ff{b(\\zeta,\\mu,\\theta)}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta,\\mu,\\theta)|}\\Big)\\\\\n&=\\ff{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)(\\zeta,\\mu,\\theta)}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta,\\mu,\\theta)|}-\\ff{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa( b\nb^*)(\\zeta,\\mu,\\theta)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)(\\zeta,\\mu,\\theta)}{|b(\\zeta,\\mu,\\theta)|(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta,\\mu,\\theta)|)^2}.\n\\end{split}\n\\end{equation}\nNext, for any random variables $\\zeta_1,\\zeta_2\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$ with\n$\\mathscr{L}_{\\zeta_1},\\mathscr{L}_{\\zeta_2}\\in\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$,\nit follows from \\eqref{v3} that\n\\begin{equation}\\label{v5}\n\\begin{split}\n&\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*\n(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*\n(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)\\|\\\\\n&=\\Big\\|\\ff{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|}-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^* (\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta) -\\ff{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa (\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta) (b\nb^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)}{(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|)^2|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|}\\Big\\|\\\\\n&=\\Big\\|\\ff{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*\n(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|}-\\ff{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*\n(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|}\\\\\n&\\quad-\\ff{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa (\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta) (b\nb^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)}{(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|)^2|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|}\\Big\\|\\\\\n&\\le\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)\n(\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)\\|\\\\\n&\\quad+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|\\cdot\\{\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb) (\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)\\|+\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)\n(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)\\|\\},\n\\end{split}\n\\end{equation}\nwhere in the last step we utilized the facts that $\\|A\\|=\\|A^*\\|$\nfor a matrix $A$ and that\n\\begin{equation*}\n\\begin{split}\n&\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta) (b\nb^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)\\|^2\\\\\n&=\\mbox{trace}\\Big(((\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta) (b\nb^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta))^*(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta) (b\nb^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)\\Big)\\\\\n&=\\mbox{trace}\\Big((b\nb^*)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta))((\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta) (b\nb^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)\\Big)\\\\\n&=\\mbox{trace}\\Big(((\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b) (\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)(b\nb^*)(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)(b b^*)^*(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)) \\Big)\\\\\n&=|b(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)|^4\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*\n(\\zeta_1,\\mathscr{L}_{\\zeta_1},\\theta)\\|^2.\n\\end{split}\n\\end{equation*}\nMoreover, from ({\\bf B2}), one has\n\\begin{equation}\\label{v4}\n\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b) (\\zeta_2,\\mathscr{L}_{\\zeta_2},\\theta)\\|\\le\nc\\Big\\{1+\\|\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_2}+\\mathbb{W}_2(\\mathscr{L}_{\\zeta_2},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big\\}.\n\\end{equation}\nNow, taking ({\\bf B2}), \\eqref{v5}, and \\eqref{v4}, in addition to\n\\eqref{r2} and \\eqref{t2}, into account yields that\n\\begin{equation*}\n\\begin{split}\n\\|\\Sigma_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\|&\\le c\\Big\\{(1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_2}+\\|X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q_2})\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\mathscr{L}_{X_t^0})\\\\\n&\\quad+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa\\Big(1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_1}+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big)\\Big(1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_2}+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big)\\Big\\}\\\\\n&\\quad\\times\\Big\\{1+\n \\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{1+q_3} +\n \\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big\\}\\times\\Big\\{1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})\\Big\\}.\n\\end{split}\n\\end{equation*}\nFor $q:=q_1\\vee q_2\\vee q_3,$ simple calculations and \\eqref{r6}\n give that\n\\begin{equation*}\n\\begin{split}\n\\|\\Sigma_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\|&\\le c\\Big\\{(1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{q})\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\ss{\\E\\|\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv}-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2}\\Big\\}\\\\\n&\\quad\\times\\Big\\{1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q)}+\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\Big\\}\\\\\n&\\quad+c\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa\\Big\\{1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q)}+\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\Big\\}^2\\\\\n&\\le c (1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+q)})\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\\\\n&\\quad+c (1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q)})\\ss{\\E\\|\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv}-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2}+c\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa(1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+q)})\\\\\n&=:\\tilde} \\def\\Ric{\\text{\\rm{Ric}}\\LL_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\tilde} \\def\\Ric{\\text{\\rm{Ric}}\\LL_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\tilde} \\def\\Ric{\\text{\\rm{Ric}}\\LL_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho).\n\\end{split}\n\\end{equation*}\nFor any $\\rho>0$, by virtue of H\\\"older's inequality, together with\n\\eqref{r6} and \\eqref{a9}, it follows that\n\\begin{equation}\\label{v6}\n\\begin{split}\n&\\P\\Big(\\int_0^T\\|\\tilde} \\def\\Ric{\\text{\\rm{Ric}}\\LL_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\ge\\rho\\Big)\\\\\n&\\le\\P\\Big(c\\int_0^T (1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{8(1+q)})\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\ge\\rho\\Big)\\\\\n&\\le\\P\\Big(c\\int_0^T (1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{9(1+q)})\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\ge\\rho\\Big)\\\\\n&\\le \\ff{c}{\\rho}\\int_0^T (1+\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{18(1+q)})\\ss{\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2} \\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\\\\n&\\rightarrow0\n\\end{split}\n\\end{equation}\nwhenever $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0.$ On the other hand,\nby means of \\eqref{r6}, and \\eqref{a9}, it follows that\n\\begin{equation}\\label{v7}\n\\begin{split}\n\\E\\tilde} \\def\\Ric{\\text{\\rm{Ric}}\\LL_2^2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\E\\tilde} \\def\\Ric{\\text{\\rm{Ric}}\\LL_3^2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)&\\le c (1+\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+q)})\\E\\|\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv}-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+c\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa(1+\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{8(1+q)})\\\\\n&\\le c(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\bb+\\vv^2+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\aa})\\\\\n&\\rightarrow0\n\\end{split}\n\\end{equation}\nas $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0.$ As a consequence, we\ninfer from \\eqref{v6} and \\eqref{v7} that\n\\begin{equation}\\label{b1}\n \\int_0^T \\|\\Sigma_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho) \\|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D t\\rightarrow0~~~~\\mbox{ in\n probability }\n\\end{equation}\nwhen $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0.$ Next, taking advantage\nof ({\\bf A2}), ({\\bf B3}), \\eqref{r2}, and \\eqref{v4} leads to\n\\begin{equation*}\n\\begin{split}\n&\\|\\Sigma_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\|^2+\\|\\Sigma_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\|^2\\\\&\\le c\\Big\\{1+\\|X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q)}\\Big\\}\\\\\n&\\quad\\times\\Big\\{(1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2q}+\\|X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2q})\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\mathscr{L}_{X_t^0})^2\\Big\\}\\\\\n&\\quad\\times (1+\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\mathbb{W}_2(\\mathscr{L}_{\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^2)\\\\\n&\\le c\\Big\\{(1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(1+q)}) \\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+\\E\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|^2_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\Big\\}\\\\\n&\\quad\\times \\Big\\{1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|^2_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\Big\\}\\\\\n&\\le c (1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{2(2+q)}) \\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle +c(1+\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2)\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|^2_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\\\\n&=:\\Xi_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)+\\Xi_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho),\n\\end{split}\n\\end{equation*}\nin which we adopted \\eqref{r6} in the last procedure. Via H\\\"older's\ninequality, we obtain from \\eqref{r6} and \\eqref{a9} that\n\\begin{equation}\\label{v8}\n \\E \\Xi_1(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le c(1+\\E\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(2+q)}) \\ss{\\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2}\\rightarrow0\n\\end{equation}\nas $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0.$ Also, by \\eqref{r6} and\n\\eqref{a9}, one has\n\\begin{equation}\\label{v9}\n \\E \\Xi_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\le c(1+\\E\\|\\bar Y_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2) \\E\\|\\bar\nY_{t_\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv-X_t^0\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2\\rightarrow0\n\\end{equation}\nprovided that $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0.$ Therefore,\n\\eqref{v8} and \\eqref{v9} lead to\n\\begin{equation}\\label{b2}\n \\E \\|\\Sigma_2(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\|^2+ \\E \\|\\Sigma_3(t,\\vv,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)\\|^2\\rightarrow0\n\\end{equation}\nif $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0.$ At last, the desired\nassertion \\eqref{v1} holds from \\eqref{b1} and \\eqref{b2}.\n\\end{proof}\n\n\n\n\\begin{lem}\\label{le3}\n Let $({\\bf A1})-({\\bf A3}), ({\\bf B1})-({\\bf B4})$, and $({\\bf\n C})$\nhold. Then\n\\begin{equation}\\label{c3}\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta)\\rightarrow\nK_0(\\theta):=K(\\theta)+I(\\theta)~~~~\\mbox{ in probability }\n\\end{equation}\nas $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$ and $\\vv\\rightarrow0$, where $I(\\cdot)$ and\n$K(\\cdot)$ are introduced in \\eqref{0z3} and \\eqref{0z2},\nrespectively.\n\\end{lem}\n\n\n\n\\begin{proof}\n From \\eqref{c1} and \\eqref{c2}, we deduce that\n\\begin{equation*}\n\\begin{split}\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta)& =2\n\\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}(\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*)(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\n\\circ\\Big(\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta)\\Big)\\\\\n&\\quad+2 \\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta P_k)(\\theta)\\\\\n& =-2 \\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}(b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*)(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\n\\circ\\Big(\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta)\\Big)\\\\\n&\\quad+2 \\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^* (\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta).\n\\end{split}\n\\end{equation*}\nFor any random variable $\\zeta\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$ with\n$\\mathscr{L}_\\zeta\\in\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$, by the chain rule, we\ninfer from \\eqref{v3} that\n\\begin{equation}\\label{e6}\n\\begin{split}\n\\Big(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}(b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*\\Big)(\\zeta,\\mathscr{L}_\\zeta,\\theta)\n&=\\bigg(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\bigg(\\ff{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb^*)}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b|}\\bigg)\\bigg)(\\zeta,\\mathscr{L}_\\zeta,\\theta)\\\\\n&\\quad-\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa\\bigg(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\bigg(\\ff{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^*)( b\nb^*)}{|b|(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b|)^2}\\bigg)\\bigg)(\\zeta,\\mathscr{L}_\\zeta,\\theta)\\\\\n&=(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)} b^*)\n(\\zeta,\\mathscr{L}_\\zeta,\\theta)-\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa\\Theta_1(\\zeta,\\mathscr{L}_\\zeta,\\theta).\n\\end{split}\n\\end{equation}\nNext, the chain rule shows that\n\\begin{equation*}\n\\begin{split}\n& \\Theta_1(\\zeta,\\mathscr{L}_\\zeta,\\theta)\n:= \\bigg(\\ff{|b|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)} b^*)}{1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b|}+\\ff{\\Big(b^*(\\ff{\\partial}{\\partial\\theta_1}b)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*,\\cdots,b^*(\\ff{\\partial}{\\partial\\theta_p}b)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*\\Big)_{p\\times pd}}{|b|(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b|)^2}\\\\\n&\\qquad\\qquad\\quad+\\ff{\\Big((\\ff{\\partial}{\\partial\\theta_1}(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^*))( b\nb^*),\\cdots,(\\ff{\\partial}{\\partial\\theta_p}(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*)( b\nb^*)\\Big)_{p\\times pd}}{|b|(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b|)^2}\\\\\n&\\qquad\\qquad\\quad+\\ff{\\Big((\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*( (\\ff{\\partial}{\\partial\\theta_1}b)\nb^*+b\\ff{\\partial}{\\partial\\theta_1}b^*),\\cdots,(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)^*(\n(\\ff{\\partial}{\\partial\\theta_p}b)\nb^*+b\\ff{\\partial}{\\partial\\theta_p}b^*)\\Big)_{p\\times pd}}{|b|(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b|)^2}\\\\\n&\\quad-\\ff{1+3\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b|}{|b|^3(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|b|)^3}\\Big(\n\\Big(b^*\\Big(\\ff{\\partial}{\\partial\\theta_1}b\\Big)\\Big)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*( b\nb^*),\\cdots,\\Big(b^*\\Big(\\ff{\\partial}{\\partial\\theta_p}b\\Big)\\Big)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)^*( b b^*)\\Big)\\bigg)_{p\\times\npd}(\\zeta,\\mathscr{L}_\\zeta,\\theta).\n\\end{split}\n\\end{equation*}\n Thanks to \\eqref{v3}, it follows\nthat\n\\begin{equation}\\label{e5}\n\\begin{split}\n&\\Big((\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})^*\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)})\\Big)(\\zeta,\\mathscr{L}_\\zeta,\\theta) = \\Big((\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb^*)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb)\\Big)(\\zeta,\\mathscr{L}_\\zeta,\\theta)- \\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\n\\aa}\\Theta_2(\\zeta,\\mathscr{L}_\\zeta,\\theta),\n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation*}\n\\begin{split}\n\\Theta_2(\\zeta,\\mathscr{L}_\\zeta,\\theta):&=\\bigg( \\ff{ (2|\nb|+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{\\aa}| b |^2)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^*) \\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)\n}{(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa| b |)^2} +\\ff{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^*) \\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)( b\n\\,\\, b^*) (\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\n b) }{|\nb |(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa|\nb |)^3} \\\\\n&\\quad+ \\ff{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^*) ( b \\, b^*) \\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb) }{| b |(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa| b |)^3} - \\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{ \\aa} \\ff{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b^*) ( b\n\\, b^*) \\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)( b \\, b^*) (\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b) }{| b\n|^2(1+\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa| b |)^4}\\bigg)(\\zeta,\\mathscr{L}_\\zeta,\\theta).\n\\end{split}\n\\end{equation*}\nThus, taking \\eqref{e5} and \\eqref{e6} into consideration yields\nthat\n\\begin{equation*}\n\\begin{split}\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta)& = -2\n\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}b^*)(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\n\\circ\\Big(\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)\\Gamma^{(\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho)}(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\theta,\\theta_0)\\Big)\\\\\n&\\quad+2 \\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\sum_{k=1}^n\\Big((\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\nb^*)\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)\\Big)(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\\\\\n&\\quad-2 \\sum_{k=1}^n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}b^*)(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\n\\circ\\Big(\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta_0)\\Big)\\\\\n&\\quad-2\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^\\aa \\sum_{k=1}^n\\Theta_1(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\n\\circ\\Big(\\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv)P_k(\\theta)\\Big)\\\\\n&\\quad- \\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho^{ 1+\\aa}\\sum_{k=1}^n\\Theta_2(\\bar\nY_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv,\\mathscr{L}_{\\bar Y_{(k-1)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho}^\\vv},\\theta)\\\\\n&=:\\sum_{i=1}^5I_i(n,\\vv).\n\\end{split}\n\\end{equation*}\nBy following the argument to derive \\eqref{t4}, we deduce from\n({\\bf A3}) that\n\\begin{equation}\\label{c6}\nI_1(n,\\vv)\\rightarrow K(\\theta) ~~\\mbox{ and\n}~~I_2(n,\\vv)\\rightarrow I(\\theta),~~~~\\mbox{ in probability }\n\\end{equation}\nas $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0$. Notice from ({\\bf A3})\nand \\eqref{q4} that\n\\begin{equation}\\label{c4}\n\\begin{split}\n\\|\\Theta_1\\|(\\zeta,\\mathscr{L}_\\zeta,\\theta)&\\le c\\Big((|b|+\n\\|\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)} b\\|+(1+3 |b|)\\|\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\nb\\|)\\|\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)} b\\|\\Big)(\\zeta,\\mathscr{L}_\\zeta,\\theta)\\\\\n&\\le c(1+\\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+q)}+\\mathcal\n{W}_2(\\mathscr{L}_\\zeta,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^4).\n\\end{split}\n\\end{equation}\nOn the other hand, owing to \\eqref{v4}, \\eqref{t2}, and \\eqref{q4},\none has\n\\begin{equation}\\label{c5}\n\\begin{split}\n\\|\\Theta_2\\|(\\zeta,\\mathscr{L}_\\zeta,\\theta)&\\le 2\\Big(|\nb|~\\|\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b\\|^2\\| \\hat\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta)\\| (1+ 2| b | )\n\\Big)(\\zeta,\\mathscr{L}_\\zeta,\\theta)\\\\\n&\\le c(1+\\|\\zeta\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^{4(1+q)}+\\mathcal\n{W}_2(\\mathscr{L}_\\zeta,\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho_{\\zeta_0})^4).\n\\end{split}\n\\end{equation}\nThus, by mimicking the argument of \\eqref{q6}, we obtain from\n\\eqref{c4} that\n\\begin{equation}\\label{c7}\nI_3(n,\\vv)\\rightarrow0~~\\mbox{ and\n}~~I_4(n,\\vv)\\rightarrow0~~~\\mbox{ in probability }\n\\end{equation}\nas $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0$. Furthermore, \\eqref{r6}\nand \\eqref{c5} enable us to get that\n\\begin{equation}\\label{c8}\nI_5(n,\\vv)\\rightarrow0 ~~~\\mbox{ in probability }\n\\end{equation}\nwhenever $\\vv\\rightarrow0$ and $\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\rightarrow0$.\n Thus, the desired\nassertion \\eqref{c3} follows from \\eqref{c6}, \\eqref{c7}, as well as\n\\eqref{c8}.\n\\end{proof}\n\n\n\n\nNow, we move forward to complete the\n\\begin{proof}[ Proof of Theorem \\ref{th2}]\nWith Lemmas \\ref{le2} and \\ref{le3} at hand, the proof of Theorem\n\\ref{th2} is parallel to that of \\cite[Theorem 4.1]{RW}. Whereas, to\nmake the content self-contained, we give an outline of the proof. In\nterms of Theorem \\ref{th1}, there exists a sequence\n$\\eta_{n,\\vv}\\rightarrow0$ as $\\vv\\rightarrow0$ and $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$\nsuch that $\\hat\\theta_{n,\\vv}\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)\\subset\\Theta$, $\\P$-a.s. By the Taylor\nexpansion, one has\n\\begin{equation}\\label{a4}\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Phi_{n,\\vv})(\\hat\\theta_{n,\\vv})=(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Phi_{n,\\vv})(\\theta_0)+D_{n,\\vv}(\\hat\\theta_{n,\\vv}-\\theta_0),~~~\\hat\\theta_{n,\\vv}\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)\n\\end{equation}\nwith\n\\begin{equation*}\nD_{n,\\vv}:=\\int_0^1(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0\n+u(\\hat\\theta_{n,\\vv}-\\theta_0))\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D u,~~~~\\hat\\theta_{n,\\vv}\\in\nB_{\\eta_{n,\\vv}}(\\theta_0).\n\\end{equation*}\n Observe that, for $\\hat\\theta_{n,\\vv}\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)$,\n\\begin{equation*}\n\\begin{split}\n\\|D_{n,\\vv}-K_0(\\theta_0)\\|&\\le\\|D_{n,\\vv}-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0)\\|+\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0)-K_0(\\theta_0)\\|\\\\\n&\\le\\int_0^1\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0\n+u(\\hat\\theta_{n,\\vv}-\\theta_0))-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0)\\|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nu\\\\\n&\\quad+ \\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0)-K_0(\\theta_0)\\|\\\\\n&\\le \\sup_{\\theta\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)}\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta)-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0)\\|+\n\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0)-K_0(\\theta_0)\\|\\\\\n&\\le\\sup_{\\theta\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)}\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta)-K_0(\\theta)\\|+\\sup_{\\theta\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)}\\|K_0(\\theta)-K_0(\\theta_0)\\|\\\\\n&\\quad+2\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta_0)-K_0(\\theta_0)\\|.\n\\end{split}\n\\end{equation*}\n This, together\nwith Lemma \\ref{le3} and continuity of $K_0(\\cdot)$, gives that\n\\begin{equation}\\label{a0}\nD_{n,\\vv}\\rightarrow K_0(\\theta_0)~~~~\\mbox{ in probability }\n\\end{equation}\nas $\\vv\\rightarrow0$ and $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle.$ By following the exact\nline of \\cite[Theorem 2.2]{LSS}, we can deduce that $D_{n,\\vv}$ is\ninvertible on the set\n\\begin{equation*}\n\\Gamma_{n,\\vv}:=\\Big\\{\\sup_{\\theta\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)}\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta)-K_0(\\theta_0)\\|\\le\\ff{\\aa}{2},~~\\hat\\theta_{n,\\vv}\\in\nB_{\\eta_{n,\\vv}}(\\theta_0) \\Big\\}\n\\end{equation*}\nfor some constant $\\aa>0.$ Let\n\\begin{equation*}\n\\mathscr{D}_{n,\\vv}=\\{D_{n,\\vv} \\mbox{ is invertible },\n\\hat\\theta_{n,\\vv}\\in B_{\\eta_{n,\\vv}}(\\theta_0) \\}.\n\\end{equation*}\nBy virtue of Lemma \\ref{le3}, one has\n\\begin{equation}\\label{n1}\n\\lim_{\\vv\\rightarrow0,n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle}\\P\\Big(\\sup_{\\theta\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)}\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta)-K_0(\\theta_0)\\|\\le\\ff{\\aa}{2}\\Big)=1.\n\\end{equation}\nOn the other hand, recall that\n\\begin{equation}\\label{n2}\n\\lim_{\\vv\\rightarrow0,n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle}\\P\\Big(\\hat\\theta_{n,\\vv}\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)\\Big)=1.\n\\end{equation}\nBy the fundamental fact: for any events $A,B$,\n$\\P(AB)=\\P(A)+\\P(B)-\\P(A\\cup B)$, we observe that\n\\begin{equation}\\label{n3}\n\\begin{split}\n1\\ge\\P(\\Gamma_{n,\\vv})&\\ge\\P\\Big(\\sup_{\\theta\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)}\\|(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta^{(2)}\\Phi_{n,\\vv})(\\theta)-K_0(\\theta_0)\\|\\le\\ff{\\aa}{2}\\Big)\\\\\n&\\quad+\\P\\Big(\\hat\\theta_{n,\\vv}\\in\nB_{\\eta_{n,\\vv}}(\\theta_0)\\Big)-1.\n\\end{split}\n\\end{equation}\nThus, taking advantage of \\eqref{n1}, \\eqref{n2} as well as\n\\eqref{n3}, we deduce from Sandwich theorem that\n\\begin{equation}\\label{n4}\n\\P(\\mathscr{D}_{n,\\vv})\\ge \\P(\\Gamma_{n,\\vv})\\rightarrow1\n\\end{equation}\n as $\\vv\\rightarrow0$ and $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$. Set\n\\begin{equation*}\nU_{n,\\vv}:=D_{n,\\vv}{\\bf1}_{\\mathscr{D}_{n,\\vv}}+I_{p\\times\np}{\\bf1}_{\\mathscr{D}_{n,\\vv}^c},\n\\end{equation*}\nwhere $I_{p\\times p}$ is a $p\\times p$ identity matrix. For\n$S_{n,\\vv}:=\\vv^{-1}(\\hat\\theta_{n,\\vv}-\\theta_0)$, we deduce from\n\\eqref{a4} that\n\\begin{align*}\nS_{n,\\vv}&=S_{n,\\vv}{\\bf1}_{\\mathscr{D}_{n,\\vv}}+ S_{n,\\vv}{\\bf1}_{\\mathscr{D}_{n,\\vv}^c}\\\\\n&=U_{n,\\vv}^{-1}D_{n,\\vv}S_{n,\\vv}{\\bf1}_{\\mathscr{D}_{n,\\vv}}+ S_{n,\\vv}{\\bf1}_{\\mathscr{D}_{n,\\vv}^c}\\\\\n&=\\vv^{-1}U_{n,\\vv}^{-1}\\{(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Phi_{n,\\vv})(\\hat\\theta_{n,\\vv})-(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Phi_{n,\\vv})(\\theta_0)\\}{\\bf1}_{\\mathscr{D}_{n,\\vv}}\n+ S_{n,\\vv}{\\bf1}_{\\mathscr{D}_{n,\\vv}^c}\\\\\n&=-\\vv^{-1}U_{n,\\vv}^{-1}(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Phi_{n,\\vv})(\\theta_0){\\bf1}_{\\mathscr{D}_{n,\\vv}}\n+ S_{n,\\vv}{\\bf1}_{\\mathscr{D}_{n,\\vv}^c}\\\\\n&\\rightarrow I^{-1}(\\theta_0)\\int_0^T\\Upsilon(X_s^0,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nB(s),\n\\end{align*}\nas $\\vv\\rightarrow0$ and $n\\rightarrow\\infty$, where in the forth\nidentity we dropped the term\n$(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta\\Phi_{n,\\vv})(\\hat\\theta_{n,\\vv})$ according to the\nnotion of LSE and Fermat's lemma, and the last display follows from\nLemma \\ref{le2}, \\eqref{a0} as well as \\eqref{n4} and by noting\n$K_0(\\theta_0)=I(\\theta_0)$. We therefore complete the proof.\n\\end{proof}\n\n\n\n\\section{Proof of Example \\ref{exa}}\n\n\\begin{proof}[Proof of Example \\ref{exa}]\nIt is sufficient to check all of assumptions in Theorems \\ref{th1}\nand Theorem \\ref{th2} are fulfilled.\n\nFor any $\\zeta,\\zeta'\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$, $\\mu\\in\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$ and\n$\\theta=(\\theta^{(1)},\\theta^{(2)})^*\\in\\Theta_0$, set\n\\begin{equation}\\label{ex1}\nb_0(\\zeta,\\zeta'):=-\\zeta^3(0)+\\zeta(0)+\\int_{-r_0}^0\\zeta(v)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nv+\\int_{-r_0}^0\\zeta'(v)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D v,\n\\end{equation}\n\\begin{equation*}\nb(\\zeta,\\mu,\\theta):=\\theta^{(1)}+\\theta^{(2)}\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\nb_0(\\zeta,\\zeta')\\mu(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta') ~~\\mbox{ and\n}~~\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta):=\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta,\\mu):=1+\\int_{-r_0}^0|\\zeta(v)|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D v.\n\\end{equation*}\nThen, \\eqref{d1} can be reformulated as \\eqref{eq1}. By \\eqref{ex1}\nand H\\\"older's inequality, we find out some constants $c_1,c_2>0$\nsuch that\n\\begin{equation}\\label{d3}\n\\begin{split}\n&\\<\\zeta_1(0)-\\zeta_2(0),b(\\zeta_1,\\mu,\\theta)-b(\\zeta_2,\\mu,\\theta)\\>\\\\\n&= \\theta^{(2)} \\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r \\<\\zeta_1(0)-\\zeta_2(0),b_0(\\zeta_1,\\zeta)-\nb_0(\\zeta_2,\\zeta)\\>\\mu_t(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta) \\\\\n&\\le c_1\\Big\\{|\\zeta_1(0)-\\zeta_2(0)|^2+\\int_{-r_0}^0|\\zeta_1(v)-\\zeta_2(v)|^2\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D v\\Big\\}\\\\\n&\\le c_2 \\,\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2,~~~~~\\mu\\in\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r),\n~~\\zeta_1,\\zeta_2 \\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\n\\end{split}\n\\end{equation}\nNext, we deduce from \\eqref{ex1} that for some constant $c_3>0,$\n\\begin{equation*}\n\\begin{split}\n|b(\\zeta,\\mu,\\theta)-b(\\zeta,\\nu,\\theta)|&\\le\\theta^{(2)}\\Big|\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\nb_0(\\zeta,\\zeta_1)\\mu(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta_1)-\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\nb_0(\\zeta,\\zeta_2)\\nu(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta_2)\\Big|\\\\\n&\\le\\theta^{(2)}\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r| b_0(\\zeta,\\zeta_1)-\nb_0(\\zeta,\\zeta_2)|\\pi(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta_1,\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta_2)\\\\\n&\\le c_3\\,\\mathbb{W}_2(\\mu,\\nu),~~~~\\zeta\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r,~~~\\mu,\\nu\\in\\mathcal\n{P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r),\n\\end{split}\n\\end{equation*}\nin which $\\pi\\in\\mathcal {C}(\\mu,\\nu)$. Therefore, ({\\bf A1}) holds\ntrue. Next,\n for any\n$\\zeta_1,\\zeta_2\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$ and $\\mu,\\nu\\in\\mathcal {P}_2(\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r)$, we obtain\nthat\n\\begin{equation*}\n |\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta_1,\\mu)-\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\zeta_2,\\nu) |\\le\\int_{-r_0}^0|\\zeta_1(\\theta)-\\zeta_2(\\theta)|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\theta\\le r_0\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle.\n\\end{equation*}\nSo ({\\bf A2}) is satisfied. For any\n$\\zeta_1,\\zeta_2,\\zeta^{(1)},\\zeta^{(2)}\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r$, note that\n\\begin{equation}\\label{ex2}\n\\begin{split}\n&|b_0(\\zeta_1,\\zeta^{(1)})-b_0(\\zeta_2,\\zeta^{(2)})|\\\\&\\le\n|\\zeta_1^3(0)-\\zeta_2^3(0)|+|\\zeta_1(0)-\\zeta_2(0)|+\\int_{-r_0}^0|\\zeta_1(v)-\\zeta_2(v)|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\nv+\\int_{-r_0}^0|\\zeta^{(1)}(v)-\\zeta^{(2)}(v)|\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D v\\\\\n&\\le\nc_4(1+\\zeta_1^2(0)+\\zeta_2^2(0))|\\zeta_1(0)-\\zeta_2(0)|+r_0\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+r_0\\|\n\\zeta^{(1)}-\\zeta^{(2)}\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\\\\\n&\\le\nc_5(1+\\|\\zeta_1\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\|\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2)\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+r_0\\|\n\\zeta^{(1)}-\\zeta^{(2)} \\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\n\\end{split}\n\\end{equation}\nfor some constants $c_4,c_5>0.$ Next, we have\n\\begin{equation}\\label{d4}\n(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b)(\\zeta,\\mu,\\theta)=\\Big(1,\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\nb_0(\\zeta,\\zeta')\\mu(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta')\\Big)^*~~~\\mbox{ and\n}~~~(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta(\\nabla} \\def\\pp{\\partial} \\def\\EE{\\scr E_\\theta b))(\\zeta,\\mu,\\theta)={\\bf 0}_{2\\times2},\n\\end{equation}\nwhere ${\\bf 0}_{2\\times2}$ stands for the $2\\times 2$-zero matrix.\nThus, \\eqref{ex2} and \\eqref{d4} enable us to deduce that ({\\bf B2})\nand ({\\bf C}) hold, respectively. Furthermore, due to \\eqref{ex2},\nwe find that\n\\begin{equation*}\n\\begin{split}\n|b(\\zeta_1,\\mu,\\theta)-b(\\zeta_2,\\nu,\\theta)|&\\le\\theta^{(2)}\\Big|\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\nb_0(\\zeta_1,\\zeta^{(1)})\\mu(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta^{(1)})-\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\nb_0(\\zeta_2,\\zeta^{(2)})\\nu(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta^{(2)})\\Big|\\\\\n&\\le\\theta^{(2)}\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r| b_0(\\zeta_1,\\zeta^{(1)})-\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\nb_0(\\zeta_2,\\zeta^{(2)})|\\pi(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta^{(1)},\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta^{(2)})\\\\\n&\\le\nc_6(1+\\|\\zeta_1\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2+\\|\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle^2)\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle+c_6\\mathbb{W}_2(\\mu,\\nu).\n\\end{split}\n\\end{equation*}\nTherefore, we infer that ({\\bf B1}) holds. Next, observe that\n\\begin{equation*}\n|\\sigma} \\def\\ess{\\text{\\rm{ess}}^{-2}(\\zeta_1,\\mu)-\\sigma} \\def\\ess{\\text{\\rm{ess}}^{-2}(\\zeta_2,\\nu)| \\le\nc_7\\|\\zeta_1-\\zeta_2\\|_\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle\n\\end{equation*}\nfor some $c_7>0.$ Consequently, ({\\bf B3}) is true.\n\n\n\n\n\n\n\n\nThe discrete-time EM scheme associated with \\eqref{d1} is given by\n\\begin{equation}\nY^\\vv(t_k)=Y^\\vv(t_{k-1})+\\Big(\\theta^{(1)}+\\theta^{(2)}\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\nb_0(\\hat Y^\\vv_{t_{k-1}},\\zeta)\\mathscr{L}_{\\hat\nY^\\vv_{t_{k-1}}}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta)\\Big)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho+\\vv\\,\\sigma} \\def\\ess{\\text{\\rm{ess}}(\\hat\nY^\\vv_{t_{k-1}})\\triangle B_k,~~~k\\ge1,\n\\end{equation}\nwith $Y^\\vv(t)=X^\\vv(t)=\\xi(t), t\\in[-r_0,0],$ where $(\\hat\nY^\\vv_{t_k})$ is defined as in \\eqref{w2}. According to \\eqref{eq2},\nthe contrast function admits the form below\n\\begin{eqnarray*}\n\\Psi_{n,\\vv}(\\theta)&=&\\vv^{-2}\\delta^{-1}\\sum_{k=1}^n\\ff{1}{(1+|Y^\\vv(t_{k-1})|)^2}\\Big|Y^\\vv(t_k)\n-Y^\\vv(t_{k-1})\\\\\n&&\\qquad\\qquad-\\Big(\\theta^{(1)}+\\theta^{(2)}\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\nb_0(\\hat Y^\\vv_{t_{k-1}},\\zeta)\\mathscr{L}_{\\hat\nY^\\vv_{t_{k-1}}}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta)\\Big)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\Big|^2.\n\\end{eqnarray*}\nObserve that\n\\begin{equation*}\n\\begin{split}\n\\ff{\\partial}{\\partial\\theta^{(1)}}\\Psi_{n,\\vv}(\\theta)&=-2\\,\\vv^{-2}\\sum_{k=1}^n\\ff{1}{(1+|Y^\\vv(t_{k-1})|)^2}\\Big\\{Y^\\vv(t_k)-Y^\\vv(t_{k-1})\\\\\n&\\quad-\\Big(\\theta^{(1)}+\\theta^{(2)}\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r b_0(\\hat\nY^\\vv_{t_{k-1}},\\zeta)\\mathscr{L}_{\\hat\nY^\\vv_{t_{k-1}}}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta)\\Big)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\Big\\},\n\\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split}\n\\ff{\\partial}{\\partial\\theta^{(2)}}\\Psi_{n,\\vv}(\\theta)&=-2\\,\\vv^{-2}\\sum_{k=1}^n\\ff{1}{(1+|Y^\\vv(t_{k-1})|)^2}\\Big\\{Y^\\vv(t_k)-Y^\\vv(t_{k-1})\\\\\n&\\quad-\\Big(\\theta^{(1)}+\\theta^{(2)}\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r b_0(\\hat\nY^\\vv_{t_{k-1}},\\zeta)\\mathscr{L}_{\\hat\nY^\\vv_{t_{k-1}}}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta)\\Big)\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho\\Big\\} \\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r b_0(\\hat\nY^\\vv_{t_{k-1}},\\zeta)\\mathscr{L}_{\\hat Y^\\vv_{t_{k-1}}}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta).\n\\end{split}\n\\end{equation*}\nSubsequently, solving the equation below\n\\begin{equation*}\n\\ff{\\partial}{\\partial\\theta^{(1)}}\\Psi_{n,\\vv}(\\theta)=\\ff{\\partial}{\\partial\\theta^{(2)}}\\Psi_{n,\\vv}(\\theta)=0,\n\\end{equation*}\nwe then obtain the LSE\n$\\hat\\theta_{n,\\vv}=(\\hat\\theta_{n,\\vv}^{(1)},\\hat\\theta_{n,\\vv}^{(2)})^*$\nof the unknown parameter\n$\\theta=(\\theta^{(1)},\\theta^{(2)})^*\\in\\Theta_0$ with the following \n\\begin{equation*}\n\\hat\\theta_{n,\\vv}^{(1)}=\\ff{A_2A_5-A_3A_4}{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho(A_1A_5-A_4^2)}~~~~~\\mbox{\nand }~~~~~\n\\hat\\theta_{n,\\vv}^{(2)}=\\ff{A_1A_3-A_2A_4}{\\delta} \\def\\DD{\\Delta} \\def\\vv{\\varepsilon} \\def\\rr{\\rho(A_1A_5-A_4^2)},\n\\end{equation*}\nwhere\n\\begin{equation*}\nA_1:=\\sum_{k=1}^n\\ff{1}{(1+|Y^\\vv(t_{k-1})|)^2},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n~~A_2:=\\sum_{k=1}^n\\ff{Y^\\vv(t_k)-Y^\\vv(t_{k-1})}{(1+|Y^\\vv(t_{k-1})|)^2},\n\\end{equation*}\n\\begin{equation*}\nA_3:=\\sum_{k=1}^n\\ff{(Y^\\vv(t_k)-Y^\\vv(t_{k-1}))\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r b_0(\\hat\nY^\\vv_{t_{k-1}},\\zeta)\\mathscr{L}_{\\hat\nY^\\vv_{t_{k-1}}}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta)}{(1+|Y^\\vv(t_{k-1})|)^2},~~~A_4:=\\sum_{k=1}^n\\ff{\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r\nb_0(\\hat Y^\\vv_{t_{k-1}},\\zeta)\\mathscr{L}_{\\hat\nY^\\vv_{t_{k-1}}}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta)}{(1+|Y^\\vv(t_{k-1})|)^2},\n\\end{equation*}\nand\n\\begin{equation*}\nA_5:=\\sum_{k=1}^n\\ff{\\Big(\\int_\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r b_0(\\hat\nY^\\vv_{t_{k-1}},\\zeta)\\mathscr{L}_{\\hat\nY^\\vv_{t_{k-1}}}(\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D\\zeta)\\Big)^2}{(1+|Y^\\vv(t_{k-1})|)^2}.\n\\end{equation*}\nIn terms of Theorem \\ref{th1}, $\\hat\\theta_{n,\\vv}\\rightarrow\\theta$\nin probability as $\\vv\\rightarrow0$ and $n\\rightarrow\\infty$. Next,\nfrom \\eqref{d4}, it follows that\n\\begin{equation*}\nI(\\theta_0)=\\left(\\begin{array}{ccc}\n \\int_0^T\\ff{1}{(1+|X_s^0|)^2}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s & \\int_0^T\\ff{b_0(X_s^0,X_s^0)}{(1+|X_s^0|)^2}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n \\int_0^T\\ff{b_0(X_s^0,X_s^0)}{(1+|X_s^0|)^2}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s & \\int_0^T\\ff{b_0^2(X_s^0,X_s^0)}{(1+|X_s^0|)^2}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D s\\\\\n \\end{array}\n \\right),\n\\end{equation*}\nand, for $\\zeta\\in\\scr C} \\def\\aaa{\\mathbf{r}} \\def\\r{r,$\n\\begin{equation*}\n\\int_0^T\\Upsilon(X_s^0,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(s)=\\left(\\begin{array}{c}\n \\int_0^T\\ff{1}{1+|X^0(s)|}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(s) \\\\\n \\int_0^T\\ff{ b_0(X_s^0,X_s^0)}{1+|X^0(s)|}\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(s)\\\\\n \\end{array}\n \\right).\n\\end{equation*}\nAt last, according to Theorem \\ref{th2}, we conclude that\n\\begin{equation*}\n\\vv^{-1}(\\hat\\theta_{n,\\vv}-\\theta_0)\\rightarrow\nI^{-1}(\\theta_0)\\int_0^T\\Upsilon(X_s^0,\\theta_0)\\text{\\rm{d}}} \\def\\bb{\\beta} \\def\\aa{\\alpha} \\def\\D{\\scr D B(s)~~~~\\mbox{ in\nprobability }\n\\end{equation*}\nas $\\vv\\rightarrow0$ and $n\\rightarrow\\infty}\\def\\X{\\mathbb{X}}\\def\\3{\\triangle$ provided that $I(\\cdot)$\nis positive definite.\n\n\n\n\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \nIt is widely believed that the cosmic microwave background (CMB) will\nbecome the premier laboratory for the study of the early universe and\nclassical cosmology. This belief relies on the high precision of the\nupcoming MAP\\footnote{{\\tt http:\/\/map.gsfc.nasa.gov}}\nand Planck Surveyor\\footnote{{\\tt http:\/\/astro.estec.eas.nl\/SA-general\/Projects\/Planck}} satellite missions and the high accuracy of theoretical\npredictions of CMB anisotropies given a definite model for\nstructure formation (\\cite{Hu95}\\ 1995). To realize the potential of the CMB,\naspects of structure formation affecting anisotropies at only the\npercent level in power must be taken into account. \n\nA great uncertainty in models for structure formation is the extent\nand nature of reionization. \nFortunately, this uncertainty is largely not reflected in\nthe CMB anisotropies due to the low optical depth to Thomson\nscattering at the low redshifts in question.\nReionization is known to be essentially\ncomplete by $z \\sim 5$ from the absence of the Gunn-Peterson effect in\nquasar absorption spectra (\\cite{Gun65}\\ 1965). Significant\nreionization before $z\\sim 50$ will be ruled out once the tentative\ndetections of the CMB acoustic peaks at present are confirmed \n(\\cite{Sco95} 1995). It should be possible to\ndeduce the reionization redshift $z_i$ through CMB polarization\nmeasurements (\\cite{Zal97}\\ 1997). \n\nNevertheless, the\nduration of time spent in a partially ionized state will remain\nuncertain. \nMoreover as emphasized by\n\\cite{Kai84} (1984) secondary anisotropies generated by the Doppler\neffect in linear perturbation theory are suppressed on small scales\nfor geometric reasons (gravitational instability generates potential flows, \nleading to cancellations between positive and negative Doppler shifts). Higher\norder effects which are not generally included in the theoretical\nmodeling of CMB anisotropies are likely to be the main source of\nsecondary anisotropies from reionization below the degree scale. Such\neffects rely on modulating the Doppler effect with spatial variations\nin the optical depth. Incarnations of this general mechanism include\nthe Vishniac effect from linear density variations (\\cite{Vis87}\\\n1987), the kinetic Sunyaev-Zel'dovich effect from clusters (\\cite{SZ}\\\n1970), and the effect considered here: the spatial variation of the\nionization fraction.\n\nReionization commences when the first baryonic objects form stars or\nquasars that convert a part of the nuclear or gravitational energy\ninto UV photons. Each such source then blows out an ionization sphere around it. Before these\nregions overlap is a period when the universe is ionized in patches.\nThe extent of this period and the time evolution of the size and\nnumber density of these patches depend on the nature of the ionizing\nengines in the first baryonic objects. Theories of reionization do not\ngive robust constraints \n({\\it cf.} \\cite{Teg94}\\ 1994; \\cite{Ree96}\\ 1996;\n\\cite{Agh96}\\ 1996; \\cite{Loe97}\\ 1997; \\cite{Hai97}\\ 1997; \\cite{Sil98}\\ 1998;\n\\cite{Hai98}\\ 1998). \n\nWe therefore take a phenomenological approach to\nstudying the effects of patchy reionization on the CMB. We introduce\na simple but illustrative three parameter model for the reionization\nprocess based on the redshift of its onset $z_i$, the duration before\ncompletion $\\delta z$, and the typical comoving size of the patches\n$R$. It is then straightforward to calculate the CMB anisotropies\ngenerated by the patchiness of the ionization degree of the\nintergalactic medium.\n\nWe find that only the most extreme models of reionization can produce\ndegree scale anisotropies that are observable in the power spectrum\n given the cosmic\nvariance limitations. A large signal on degree scales requires early\nionization, $z_i\\gtrsim 30$, long duration, $\\delta z\\sim z_i$, and\nionization in very large patches, $R\\gtrsim 30$Mpc. Thus the\npatchiness of reionization is unlikely to affect cosmological\nparameter estimation from the acoustic peaks in the CMB (\\cite{Jun96}\\\n1996;\n\\cite{Zal97}\\ 1997; \\cite{Bon97}\\ 1997).\n \nOn the other hand, the patchy reionization signal on the sub-arcminute\nscale can, in principle, surpass both the primary and the secondary Vishniac\nsignals. These may be detectable by the Planck Surveyor and upcoming \nradio interferometry measurements (\\cite{Par97}\\ 1997) \nif point sources can be removed\nat the $\\Delta T\/T \\sim 10^{-6}$ level. \n\nAn explicit expression for the CMB anisotropies power spectrum\ngenerated in a universe reionized in patches is given \\S \\ref{sec:explicit}. \nSimple\norder of magnitude estimate of the anisotropy from patches is given in\n\\S \\ref{sec:order}. In \\S \\ref{sec:power} we give a rigorous definition of our three-parameter\nreionization model, and calculate the patchy part of the power\nspectrum. We discuss illustrative examples in \\S \\ref{sec:discussion}.\n\n\n\\section{CMB power spectrum}\n\\subsection{Explicit expression}\n\\label{sec:explicit}\nTemperature perturbations $\\Delta \\equiv \\delta T\/T$ are generated by\nDoppler shifts from Thomson scattering. For small optical depths\n\\begin{equation}\n\\Delta =-\\int d{\\bf l}\\cdot {\\bf v}\\sigma _Tnx_e.\n\\label{eqn:fundamental}\n\\end{equation}\nAll quantities here are in physical units. The integral is along the line\nof site, ${\\bf v}$ is the peculiar velocity of matter, $c=1$,\n$\\sigma_T$ is the Thomson cross section, $n$ is the number density of\nfree and bound electrons, $x_e$ is the local ionization fraction. \n\nTo evaluate equation~(\\ref{eqn:fundamental}) explicitly one must specify\nthe cosmological model. For simplicity, we take a universe\nwith critical density in matter throughout; we describe the generalization\nto an arbitrary FRW universe in \\S \\ref{sec:power}. \nWe furthermore use comoving coordinates ${\\bf x}$ and conformal time\n$\\eta\\equiv \\int (1+z)dt= (1+z)^{-1\/2} \\eta_0$, \nwhere $\\eta_0=2\/H_0$ is the present particle horizon. \nWe observe at ${\\bf x}=0$ and conformal time $\\eta_0$\nalong the direction of a unit vector $\\hat{\\gamma}$; light propagation\nis given by ${\\bf x}=\\hat{\\gamma}(\\eta _0-\\eta )$. \nEquation~(\\ref{eqn:fundamental})\ncan be\nwritten as\n\\begin{equation}\n\\Delta (\\hat{\\gamma})=-\\tau_0 \\eta_0^3\n\\int {d\\eta \\over \\eta^4} \\hat{\\gamma}\\cdot {\\bf v}[ \\eta ,\\hat{\\gamma}(\\eta _0-\\eta )] x_e[ \\eta ,\\hat{\\gamma}(\\eta _0-\\eta )] .\n\\end{equation}\nHere $\\tau _0\\equiv \\sigma _Tn_0 \\eta_0$ is\nthe optical depth to Thomson scattering across the present particle horizon.\n\nThe\nscales contributing to the peculiar velocity are still in the linear\nregime, therefore\n\\begin{equation}\n{\\bf v}(\\eta ,{\\bf x})={\\eta \\over \\eta _0}{\\bf v}({\\bf x}),\n\\end{equation}\nwhere ${\\bf v}({\\bf x})$ is the peculiar velocity today. The final\nexplicit expression for the CMB temperature perturbation generated\nduring reionization is\n\\begin{equation}\n\\Delta (\\hat{\\gamma})=-\\tau_0 \\eta_0^2\n\\int {d\\eta \\over \\eta ^3} \\hat{\\gamma}\\cdot {\\bf v}[ \\hat{\\gamma}(\\eta _0-\\eta )] x_e[ \\eta ,\\hat{\\gamma}(\\eta _0-\\eta )] .\n\\label{eqn:temperature}\n\\end{equation}\nThe correlation function of the temperature perturbations is defined\nas\n\\begin{equation}\nC(\\theta )=\\left<\\Delta (\\hat{\\gamma}_1)\\Delta\n(\\hat{\\gamma}_2)\\right>|_{\\hat{\\gamma}_1\\cdot \\hat{\\gamma}_2=\\cos \\theta}.\n\\end{equation}\nWith temperature perturbations given by equation~(\\ref{eqn:temperature}), \nthis becomes\n\\begin{eqnarray}\nC(\\theta )&=&\\tau_0^2 \\eta _0^4 \\int {d\\eta _1\\over \\eta _1^3}\n\\int {d\\eta _2\\over \\eta _2^3} \\big<\n\\hat{\\gamma}_1\\cdot {\\bf v}({\\bf x}_1)\n\\hat{\\gamma}_2\\cdot\n{\\bf v}({\\bf x}_2)\n\\nonumber\\\\ &&\\quad \\times \\, x_e(\\eta _1,{\\bf x}_1)x_e(\\eta _2,{\\bf x}_2)\\big>,\n\\label{eqn:correlationgeneral}\n\\end{eqnarray}\nwhere we denote ${\\bf x}_1\\equiv \\hat{\\gamma}_1(\\eta _0-\\eta _1)$ and\n${\\bf x}_2\\equiv \\hat{\\gamma}_2(\\eta _0-\\eta _2)$.\n\n\\subsection{Order of magnitude estimates}\n\\label{sec:order}\n\nConsider the following patchy reionization scenario. The universe was\nreionized in randomly distributed patches with a characteristic\ncomoving size $R$. The patches appeared at random in space and\ntime. Once a reionized patch appears, it moves with matter. The\naverage ionization fraction, that is the filling fraction of fully\nionized patches, grows monotonically from $X_e=0$ at high redshifts to\n$X_e=1$ at low redshifts. We consider late reionization (optical depth\nto Thomson scattering is small) and small patches (smaller than the\ncharacteristic length scale of the peculiar velocity field). We assume\nthat reionization occurred at redshift $z_i$, and the patchy phase\nduration is given by $\\delta z$.\n\nThe angular scale of the patchy signal is given by the ratio of the\nsize of patches to the distance to them in comoving coordinates, \n$\\theta \\sim R\/(\\eta_0-\\eta_i)$. \nAssuming that the patches are uncorrelated, the spectrum of \nfluctuations should be white noise above this scale which \nagrees with the exact result as we shall see (eq.~[\\ref{eqn:Cl}]). \n\nThe rms CMB temperature fluctuation $\\Delta$ on scales $\\theta$\ndue to the patchiness can be estimated as follows. Since, by\nassumption, different patches are independent, $\\Delta \\sim\nN^{1\/2}\\Delta _p$. Here $N$ is the number of patches on a line of\nsite, $\\Delta _p$ is a temperature fluctuation from one patch, $\\Delta\n_p\\sim \\tau _pv(z_i)$. Here $v(z)=(1+z)^{-1\/2}v(0)$ is the rms peculiar\nvelocity at redshift $z$, and $\\tau _p$ is the optical depth for one\npatch, $\\tau _p \\sim (1+z_i)^2\\sigma _Tn_0R$. The number of patches\n$N\\sim \\delta \\eta\/R\\sim (1+z_i)^{-3\/2}\\delta z\\eta _0\/R$. Collecting\nall the factors, we get the following estimate for the rms\nanisotropies from patches\n\\begin{equation}\n\\Delta \\sim \\tau_0\\left^{1\/2}({R\/\\eta _0})^{1\/2}(1+z_i)^{3\/4} (\\delta z)^{1\/2},\n\\end{equation}\nwhich again \nagrees with the exact result (eq.~[\\ref{eqn:amplitude}], up to a dimensionless\nmultiplier). \n\n\n\\subsection{Power spectrum}\n\\label{sec:power}\n\nWe can factor the general expression for the temperature correlation\n(\\ref{eqn:correlationgeneral}) as\n\\begin{eqnarray}\nC(\\theta )&=&\\tau_0^2 \\eta_0^4 \\int {d\\eta _1\\over \\eta _1^3}\n\\int {d\\eta _2\\over \\eta _2^3}\n\\left<\\hat{\\gamma}_1\\cdot {\\bf v}({\\bf x}_1)\\hat{\\gamma}_2\\cdot {\\bf\nv}({\\bf x}_2)\\right>\n\\nonumber \\\\\n&&\\quad\\times\\,\n\\left.\n\\end{eqnarray}\nThis assumes that $x_e$ and ${\\bf v}$ are independent random\nfields. This is not strictly correct. The ionization fraction $x_e$\nmust be determined by the density perturbation $\\delta$, and the\ndensity perturbation is not independent of the peculiar velocity (for\nexample in the linear regime $\\delta =-{1\\over 2} \\eta \\nabla \\cdot\n{\\bf v}$). However, the ionizing radiation is presumably coming from\nstrongly nonlinear objects, where first stars or quasars are\nlightening up. At high $z$, the length scales where the density is\nnonlinear are $\\ll 10$Mpc comoving, which is much smaller than the\nlength scales contributing to the peculiar velocity. Under the\nassumption of scale separation, velocity and density (and hence $x_e$)\nare indeed independent.\n\nThe correlation function for the local ionization fraction $\\left$\nis not known. Our model parameterizes the correlation function\nthrough the the patch size $R$ and \na mean (cosmic time - dependent) ionization\nfraction $X_e(\\eta )$,\n\\begin{eqnarray}\n&&\\left = X_e(\\eta _1)X_e(\\eta\n_2)\\nonumber\\\\ \n&& \\qquad + [ X_e(\\eta _{\\rm\nmin})-X_e(\\eta _1)X_e(\\eta _2)] e^{-{({\\bf x}_1-{\\bf x}_2)^2\\over\n2R^2}}\\qquad.\n\\label{eqn:xexemodel}\n\\end{eqnarray} \nHere $\\eta _{\\rm min}={\\rm min}(\\eta _1,\\eta _2)$. The Gaussian\nfunction is chosen for simplicity; it could have been any function of\nthe separation $x$ which equals 1 at $x=0$ and gradually turns to zero\nat $x>R$. For the mean ionization fraction $X_e$ we assume a change\nfrom 0 to 1 at a redshift $z_i$, with the transition occurring in a\nredshift interval $\\delta z$. We also assume $\\delta z\\ll\nz_i$ (this is true in all of the models of reionization that we are aware of).\n\n\nCMB anisotropies generated (and erased) due to the spatially constant\npart of the correlation function (\\ref{eqn:xexemodel}) are obviously the same as in the\nmodel with a uniform time-dependent reionization. The spatially\nvarying part is responsible for generating new anisotropies; its\ncontribution to erasing the primary anisotropies is negligible. The\nanisotropy suppression is mainly determined by the total optical depth\nto Thomson scattering and is insensitive to the small-scale structure\nof the ionization fraction $x_e(\\eta ,{\\bf x})$.\n\nThe CMB correlation function due to the patchy part only is\n\\begin{eqnarray}\n\\label{eqn:cthetaintegral}\nC^{\\rm (p)}(\\theta ) &=& \\tau_0^2 \\eta_0^4\\int_{0}^{\\eta _0} {d\\eta\n_1\\over \\eta _1^3} \\int_{0}^{\\eta _0} {d\\eta _2\\over \\eta\n_2^3}I_{12} \\\\ && \\quad\\times \\left<\\hat{\\gamma}_1\\cdot {\\bf\nv}({\\bf x}_1)\\hat{\\gamma}_2\\cdot {\\bf v}({\\bf x}_2)\\right>e^{-{({\\bf\nx}_1-{\\bf x}_2)^2\\over 2R^2}} \\nonumber,\n\\end{eqnarray}\nwhere we denote $I_{12}\\equiv X_e(\\eta _{\\rm min})-X_e(\\eta\n_1)X_e(\\eta _2)$. The correlation function is non-negligible only for\n$|{\\bf x}_1-{\\bf x}_2|\\lesssim R$. By assumption, $R$ is much smaller\nthen the characteristic scale of the peculiar velocity field. Also\n$|{\\bf x}_1-{\\bf x}_2|\\lesssim R$ requires that the lines of sight\n$\\hat{\\gamma}_1$ and $\\hat{\\gamma}_2$ be nearly parallel. Then\n\\begin{equation}\n\\left<\\hat{\\gamma}_1\\cdot {\\bf v}({\\bf x}_1)\\hat{\\gamma}_2\\cdot {\\bf\nv}({\\bf x}_2)\\right>\\approx {1\\over 3}\\left,\n\\end{equation}\nwhere $\\left$ is the mean squared peculiar velocity today. For $z_i\\gg\n1$, the integral (\\ref{eqn:cthetaintegral}) is dominated by small conformal times, and we\nhave $|{\\bf x}_1-{\\bf x}_2|^2\\approx \\theta ^2(\\eta _0-\\eta_i)^2+(\\eta _1-\\eta\n_2)^2$. Then\n\\begin{eqnarray}\nC^{\\rm (p)}(\\theta )&\\approx& {1\\over 3}\\tau_0^2\\eta_0^4\\lefte^{-{(\\eta\n_0 - \\eta_i)^2\\theta ^2\\over 2R^2}}\\nonumber \\\\\n&&\\times \\int_{0}^{\\infty }{d\\eta _1\\over \\eta _1^3}\n\\int_{0}^{\\infty } {d\\eta _2\\over \\eta _2^3}I_{12}e^{-{(\\eta _2-\\eta\n_1)^2\\over 2R^2}}.\n\\end{eqnarray}\nWe assume that $\\eta _i\\delta z\/(1+z_i)\\gg R$ (with $\\eta\n_i\\equiv \\eta (z_i)$) and that during the patchy phase,\n$z_i>z>z_i-\\delta z$, the ionization fraction $X_e$ grows linearly from\n0 to 1 such that eventually both hydrogen and helium are fully ionized. \nThen\n\\begin{equation}\nC^{\\rm (p)}(\\theta )=Ae^{-{\\theta ^2\\over 2\\theta _0^2}},\n\\label{eqn:cthetagen}\n\\end{equation}\nwhere the characteristic angular scale is\n\\begin{equation}\n\\theta _0={R\\over \\eta _0 - \\eta_i} = {R \\over \\eta_0} {(1+z_i)^{1\/2} \\over\n(1+z_i)^{1\/2} - 1},\n\\label{eqn:angularscale}\n\\end{equation}\nand the amplitude is\n\\begin{equation}\nA={\\sqrt{2\\pi }\\over 36} \\tau _0^2\\left{R\\over \\eta _0}\\delta\nz(1+z_i)^{3\/2}.\n\\label{eqn:amplitude}\n\\end{equation}\nNote that a critical matter-dominated universe is assumed in this\nexpression. To generalize this result replace $\\eta_0-\\eta_i$ by\nthe comoving angular diameter distance in \nequation~(\\ref{eqn:angularscale}) and a factor of $(1+z_i)$ in\nequation~(\\ref{eqn:amplitude}) with the appropriate velocity growth\nfactor.\n\nThe power spectrum is given by the spherical harmonics decomposition\n\\begin{equation}\nC_l^{\\rm (p)}=2\\pi \\int d\\cos \\theta P_l(\\cos \\theta )w_p(\\theta ).\n\\end{equation}\nFor $l\\gg 1$,\n\\begin{equation}\nC_l^{\\rm (p)}\\approx 2\\pi \\int_0^{\\infty} \\theta d\\theta J_0(l\\theta )w_p(\\theta\n)=2\\pi A\\theta _0^2e^{-{\\theta _0^2l^2\\over 2}}.\n\\end{equation}\nThe power per octave is\n\\begin{equation}\n{l^2C_l^{\\rm (p)}\\over 2\\pi }=Al^2\\theta _0^2e^{-{\\theta _0^2l^2\\over 2}}.\n\\label{eqn:Cl}\n\\end{equation}\n\nThe anisotropy power reaches the maximal value\n\\begin{equation}\n({l^2C_l^{\\rm (p)}\\over 2\\pi })_{\\rm max}={\\sqrt{2\\pi }\\over 18{\\rm e}} \\tau\n_0^2\\left{R\\over \\eta _0}\\delta z(1+z_i)^{3\/2},\n\\end{equation}\nat\n\\begin{equation}\nl_{\\rm max}={\\sqrt{2} \\eta _0\\over R} [1 - (1+z_i)^{-1\/2}].\n\\end{equation}\n\n\n\\subsection{Discussion}\n\\label{sec:discussion}\n\nThe signal from patchy reionization in our model depends on four\nquantities: the rms peculiar velocity \n$\\left< v^2 \\right>^{1\/2}$ today, the redshift of reionization $z_i$,\nits duration $\\delta z$ and the characteristic comoving size of the patches $R$.\nThe structure formation model specifies\nthe power spectrum of fluctuations which in turn tells us the \nrms peculiar velocity. Let us now consider the patchy reionized signal in \nthe context of a specific model for structure formation. \n\nFor illustrative purposes, let us consider\na cold dark matter model with $h=0.5$, $\\Omega_b =0.1$, and a scale-invariant $n=1$ \nspectrum of initial fluctuations. Normalizing the spectrum to the\nCOBE detection via the fitting formulae of \\cite{Bun97} (1997) (their\nequations [17]-[20]) and employing the analytic fit to the transfer function \nof \\cite{Eis98} (1998) (their equations [15]-[24])\nwe find an rms velocity \nof $\\left^{1\/2}=3.9\\times 10^{-3}$. \nWith the present optical depth of $\\tau_0 = 0.122 \\Omega_b h = 0.0061$,\nwe have a maximal anisotropy of\n\\begin{equation}\n({l^2C_l\\over 2\\pi })_{\\rm max}=2.41\\times 10^{-15}{R\\over {\\rm\nMpc}}\\delta z(1+z_i)^{3\/2},\n\\end{equation}\nat\n\\begin{equation}\nl_{\\rm max}={16958\\over R\/{\\rm Mpc}} [ 1 - (1+z_i)^{-1\/2}].\n\\end{equation}\n\nThe power spectrum of the model in principle also tells us the\nremaining parameters of the ionization: its redshift $z_i$,\nduration $\\delta z$ and typical patch size $R$.\nUnfortunately, these quantities depend on details of the cooling and\nfragmenting of the first baryonic objects to form the ionizing \nengines. We therefore consider $5 \\lesssim z_i \\lesssim 50$ which spans \nthe range of estimates in the literature\n(\\cite{Teg94}\\ 1994; \\cite{Ree96}\\ 1996;\n\\cite{Agh96}\\ 1996; \\cite{Loe97}\\ 1997; \\cite{Hai97}\\ 1997; \\cite{Sil98}\\ 1998;\n\\cite{Hai98}\\ 1998). Reionization, once it commences, is generally completed in\na time short compared with the expansion time at that epoch $\\delta z \/(1 +z_i) \n< 1$ by the coalescence of patches that are small compared with the \nhorizon at the time $R\/\\eta_i\n\\ll 1$ at the time. Again the exact relations depend on the efficiency with\nwhich the first objects form and create ionizing radiation \n(see e.g. \\cite{Teg94}\\ 1994). \n\n\\begin{figure}[htb]\n\\psfig{figure=patchf1.eps,width=3.3in}\n\\caption{CMB anisotropy power spectra in a CDM model with extreme patchiness. \nShown here are the primary anisotropy \nand the patchy reionization anisotropy, eq.~(\\protect\\ref{eqn:Cl}) with\n$z_i=10$, $\\delta z=3$, $R=20$Mpc. These signals are compared with\nthe cosmic variance of the primary anisotropy and the noise of\nthe MAP satellite (in logarithmic bins).}\n\\label{fig:patch10}\n\\end{figure}\n\n\nLet us consider an extreme example of $z_i=10$, $\\delta z=3$, $R=20{\\rm Mpc}$. \nThen the maximal\npower is $\\approx 5.3\\times 10^{-12}$ at $l\\approx 590$, the primary\nsignal at these scales is $\\approx 3\\times 10^{-10}$ -- the\ncontribution of the patchy reionization is small in comparison \n(see Fig.~\\ref{fig:patch10}). However, in light of \nthe high precision measurements expected from the MAP and Planck satellites \nsuch a signal is not necessarily negligible. \nThe ultimate limit of detectability through\npower spectrum measurements is provided by so-called cosmic variance.\nThis arises since we can only measure $2\\ell+1$ realizations\nof any given multipole such that power spectrum estimates will vary by\n\\begin{equation}\n\\delta C_\\ell = \\sqrt{2 \\over 2\\ell+1} C_\\ell^{(\\rm primary)}.\n\\end{equation}\nDetection of a broad feature such as that from patchy reionization is assisted\nin that we may reduce the cosmic variance by averaging over many $\\ell$'s. \nWe show an example of this averaging in Fig.~\\ref{fig:patch10} (lower left boxes).\nIn this model, the patchy reionization signal can be detected at the several\n$\\sigma$ level if cosmic variance were the main source of uncertainty. \nOf course a realistic experiment also has noise and systematic errors.\nWe also show the noise error contributions expected from the MAP experiment\nin Fig.~\\ref{fig:patch10}. \n\nAn important additional source of uncertainty is provided by other\nunknown aspects of the model. Indeed it is hoped that the CMB power spectrum\ncan be used to measure fundamental cosmological parameters to high precision.\nExcess variance from patchy reionization can in principle cause problems for\ncosmological parameter estimation from the CMB if not included in the model.\nIt would remain undetected and produce parameter misestimates if\nits signal can be accurately mimicked by variations in the other parameters.\nFortunately, the angular signature we find here -- $\\ell^2$ white noise until some\ncut off due to the patch size -- does not resemble the signature of other cosmological\nparameters which alter the positions and amplitudes of the acoustic peaks \n(see \\cite{Bon97}\\ 1997; \\cite{Zal97}\\ 1997). Coupled with the small amplitude of the effect on the 10 arcminute to degree scale for even this extreme model, it is unlikely that patchy reionization will significantly affect parameter\nestimation through the CMB. \n\nWe have called the ($z_i=10,\\delta z=3,R=20$) model extreme, because of the size of patches; the reionization redshift and duration would be considered reasonable by a number of theories. For example the early quasar model of Haiman \\& Loeb (1998) does predict $z_i\\sim 10$ and $\\delta z \\sim 3$. However, their ``medium quasar'' emits only $\\sim 10^{67}$ ionizing photons during its life time. These photons cannot ionize a bubble larger than $R\\sim 1$Mpc comoving.\n\nPerhaps more interesting is the case where reionization takes \nplace at a higher redshift with\nfor example \n$z_i=30$, $\\delta z=5$, $R=3{\\rm Mpc}$. The reduction in the patch size\ncauses the signature to move to smaller angles where the primary signal\nis negligible due to dissipational effects at recombination. \nThe increase in the optical depth at this higher redshift is counterbalanced\nby the reduction in the rms fluctuation due to the number of patches along\nthe line of site such that the amplitude of the signal increases\nonly moderately. Here the maximal \npower is $\\approx 6.2\\times10^{-12}$ at $l\\approx 4650$ \n(see Fig.~\\ref{fig:patch30}). Patchy reionization effects exceed \nthe Vishniac signal at these\nscales ($\\approx 3\\times 10^{-12}$) which is believed to be\nthe leading other source of secondary anisotropies \n(\\cite{Hu96}\\ 1996). \n\n\\begin{figure}[htb]\n\\psfig{figure=patchf2.eps,width=3.3in}\n\\caption{CMB anisotropy power spectra in a CDM model with early reionization. \nShown here are the primary anisotropy suppressed by rescattering \nand the patchy reionization anisotropy, eq.~(\\protect\\ref{eqn:Cl}) with\n$z_i=30$, $\\delta z=5$, $R=3$Mpc. These signals are compared with\nthe cosmic variance of the primary anisotropy achievable by an ideal\nexperiment in the absence of galactic and extragalactic foregrounds.}\n\\label{fig:patch30}\n\\end{figure}\n\nAlthough the morphology and amplitude \nof the patchy reionization and Vishniac signals are similar, the Vishniac\neffect is fully specified by the ionization redshift and the spectrum\nof initial fluctuations and hence may be removed once these are determined\nfrom parameter estimation at larger angular scales. Likewise, \nsince the rms peculiar velocity $\\left< v \\right>$ and the ionization\nredshift $z_i$ will be specified by the large scale observations, \nthe amplitude of the signal can be used to estimate\nthe duration of reionization $\\delta z$ and its angular location \nthe typical comoving size of the bubbles $R$. \n\nIn summary, the patchiness of reionization leaves a potentially observable \nimprint on the CMB power spectrum, but one that is unlikely to affect cosmological\nparameter estimation from the acoustic peaks in the CMB. We show how the signature\nscales with the gross properties of reionization -- its redshift, duration, and\ntypical patch size. Observational detection of this signature would provide \nuseful constraints on the presently highly uncertain reionization scenarios but\nwill likely require experiments with angular resolution of an \narcminute or better and foreground subtraction at better than the \n$\\delta T\/T \\sim 10^{-6}$ level.\n\n\\acknowledgements\nWe thank \nR. Juszkiewicz, A. Liddle, M. Tegmark and M. White\nfor discussions. This work\nwas supported by NSF PHY-9513835. WH was also supported by the\nW. M. Keck Foundation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe fascinating beauty of the theory of the functions of complex variable reveals\nitself, for example, in the harmony of the algebraic fractals on the Euclidian\nplane. It makes many researches look for the analogous number systems, the elements\nof which could be correlated not to the points on the plane but to the points of the\n4-dimensional space-time. In case of the success of such a search, we could really\ntrust the famous Pythagoras saying 'all the existing is number'. On this way, the\ninteresting results were obtained for quaternions [1], biquaternions [2-4], octaves\n[5] and so forth. Nevertheless, none of these number system theories can be compared\neven to the theory of the relatively simple 2-component complex numbers. The main\nreason for this seems to be the lack of the commutativity (and sometimes even of the\nassociativity) of the multiplication in these algebras. Although the authors of this\npaper realize the conceptual bases of all the variety of algebras, the commutativity\nof the multiplication is the integral property of all the principal number systems\nthat contain natural, integer, rational, real and complex numbers. Finally, the\ncommutativity and the associativity of the multiplication are among the axioms of\narithmetic which presents the foundation of mathematics, and it would be strange if\nthe algebraic system which is the most natural for our real world does not\ncorrespond to the rules of regular counting.\n\nOne of the systems free from this drawback is the algebra of the commutative and\nassociative hyper complex numbers, related to the direct sum of the four real\nalgebras, which will be denoted as $H_4$. The algebra of these numbers is isomorphic\nto the algebra of the 4-dimensional square real diagonal matrices, and the\ncorresponding space is a linear Finsler space with the Berwald-Moor metric (the last\nfact was proved by the authors in [6]). It should be mentioned that Finsler space\nwith the Berwald-Moor metric has been known and partially investigated for a long\ntime [7--8].\n\nOne of the main properties of this space is the existence of such a range of the\nparameters that the 3-dimensional distances (from the point of view of the observer\nwho uses the radar method to measure them [9]) correspond to the positively defined\nmetric function the limit of which is the quadratic form [10]. In other words, the\n3-dimensional world observed by an \"$H_4$ inhabitant\" is Euclidian within certain\naccuracy. Moreover, when one passes to the relativistic velocities, the\n4-dimensional intervals between the $H_4$ events present the Minkowski space\ncorrelations [11]. All this makes possible to suggest that the $H_4$ space and the\ncorresponding Finsler geometry can be used as a mathematical model of the real\nspace-time, and maybe this model would be even more productive than the pseudo\nRiemannian constructions prevailing in Physics now.\n\n\\mes\n\nAny hyper complex algebra is completely defined by the multiplication rule for the\nelements of a certain fixed basis. In the H4 number system there is a special --\nisotropic -- basis $e_1, e_2, e_3, e_4$, such that\n \\begin{equation}\\label{gp1}\n e_ie_j=p^k_{ij}e_k \\, \\qquad p^k_{ij} = \\left\\{ \\begin{array}{l}\n 1\\, , \\hbox{\\q if~} \\; i=j=k \\, , \\\\\n 0\\, , \\hbox{\\q else} \\, .\n \\end{array}\n \\right.\n \\end{equation}\n Any analytical function in this basis can be given as\n \\begin{equation}\\label{gp2}\n F(X) = f^1(\\xi^1)e_1 + f^2(\\xi^2)e_2 + f^3(\\xi^3)e_3 + f^4(\\xi^4)e_4 \\, ,\n \\end{equation}\nwhere\n \\begin{equation}\\label{gp3}\n H_4 \\ni X = \\xi^1e_1 + \\xi^2e_2 + \\xi^3e_3 + \\xi^4e_4 \\, ,\n \\end{equation}\nand $f^i$ are four arbitrary smooth functions of a single real variable.\n\nIn $H_4$ there is one more -- orthogonal -- selected basis $1, \\, j, \\, k, \\, jk$,\nwhich is related to the isotropic basis by the following formulas\n \\begin{equation}\\label{gp4}\n \\left. \\begin{array}{l}\n 1 = e_1 + e_2 + e_3 + e_4 \\, , \\\\[2pt]\n j = e_1 + e_2 - e_3 - e_4 \\, , \\\\[2pt]\n k = e_1 - e_2 + e_3 - e_4 \\, , \\\\[2pt]\n jk = e_1 - e_2 - e_3 + e_4 \\, ,\n \\end{array}\n \\right\\}\n \\end{equation}\nwhere {\\it 1} is the unity of algebra, and the corresponding component of the\nanalytical function of the $H_4$ variable is defined by the formula\n \\begin{equation}\\label{gp5}\n u = \\frac{1}{4} \\left[ f^1(\\xi^1) + f^2(\\xi^2) + f^3(\\xi^3) + f^4(\\xi^4) \\right] \\, .\n \\end{equation}\n\nIf $X$ is a radius vector, then the coordinate space $\\xi^1, \\, \\xi^2, \\, \\xi^3, \\,\n\\xi^4$ is a Berwald-Moor space with the length element\n \\begin{equation}\\label{gp6}\n ds = \\sqrt[4]{d\\xi^1d\\xi^2d\\xi^3d\\xi^4} \\equiv\n \\sqrt[4]{g_{ijkl}d\\xi^id\\xi^jd\\xi^kd\\xi^l} \\, ,\n \\end{equation}\nwhere\n \\begin{equation} \\label{gp7}\n g_{ijkl} = \\left\\{\n \\begin{array}{l}\n \\frac{1}{4!} \\, , {~~(i\\ne j\\ne k\\ne l)} , \\\\[9pt]\n \\; 0\\, \\, , {~~(else)} .\n \\end{array} \\right.\n \\end{equation}\nFor this geometry the tangent indicatrix equation is\n \\begin{equation}\\label{gp8}\n g^{ijkl}p_ip_jp_kp_l - 1 = 0 \\, ,\n \\end{equation}\nwhere\n \\begin{equation}\\label{gp9}\n g^{ijkl} = \\left\\{ \\begin{array}{l}\n \\displaystyle\\frac{4^4}{4!} \\, , {~~(i\\ne j\\ne k\\ne l)} , \\\\[9pt]\n \\; 0\\, \\, , \\hbox{~~(else)} ,\n \\end{array} \\right.\n \\end{equation}\n \\begin{equation}\\label{gp10}\n p_i = \\frac{g_{ijkl}d\\xi^jd\\xi^kd\\xi^l}{\\left(g_{mrst}d\\xi^md\\xi^rd\\xi^sd\\xi^t\n \\right)^{3\/4}} \\,\n \\end{equation}\nare the components of the generalized momentum or generalized momenta.\n\nIf we have tensors $p^k_{ij}$, $g_{ijkl}$, $g^{ijkl}$ and vector fields of the\nanalytical functions $F_{(A)}(X)$ of the $H_4$ variables, we could construct the\nmetric tensors in the 4-dimensional space-time in many ways. For example,\n \\begin{equation}\\label{gp11}\n g_{ij}(\\xi) = g_{ijkl}f^k_{(1)}f^l_{(2)} \\, ,\n \\end{equation}\nNow one can investigate the obtained Riemannian geometry. The main drawback of this\napproach is the variety of the ways to construct it.\n\n It is known [12] that if the tangent indicatrix equation is defined as\n \\begin{equation}\\label{gp12}\n \\Phi(p;\\xi) = 0 \\, ,\n \\end{equation}\nthen the geodesics will be the solutions of the canonical system of differential\nequations\n \\begin{equation}\\label{gp13}\n \\dot{\\xi}^i = \\frac{\\partial \\Phi}{\\partial p_i} \\cdot \\lambda(p;\\xi) \\, , \\qquad \\dot{p}_i\n = - \\frac{\\partial \\Phi}{\\partial \\xi^i} \\cdot \\lambda(p;\\xi) \\, ,\n \\end{equation}\n $\\lambda (p;\\xi )\\neq 0$ is an arbitrary smooth function, and a dot above $\\xi^i$ and $p_i$ means the derivation by the evolution parameter, $\\tau$.\n\n\\section{Construction of the metric function\\\\ of the pseudo Riemannian space}\n\nLet us regard a space which is conformally connected to the $H_4$ space, that is to\nthe space with the length element\n \\begin{equation}\\label{gp14}\n ds' = \\kappa(\\xi)\\cdot \\sqrt[4]{g_{ijkl}d\\xi^id\\xi^jd\\xi^kd\\xi^l} \\, ,\n \\end{equation}\nwhere $\\kappa(\\xi) > 0$ is a scalar function which is a contraction-extension\ncoefficient depending on the point.\n\nLet there be a normal congruence of geodesics (world lines). Then there is a scalar\nfunction $S(\\xi)$ (see, e.g. [12]) such that its level hyper surfaces are\ntransversal to this normal congruence of the world lines and this function is a\nsolution of the equation\n \\begin{equation}\\label{gp15}\n g^{ijkl}\\frac{\\partial S}{\\partial \\xi^i}\\frac{\\partial S}{\\partial\n \\xi^j}\\frac{\\partial S}{\\partial \\xi^k}\\frac{\\partial S}{\\partial\n \\xi^l} = \\kappa(\\xi)^4 \\, ,\n \\end{equation}\nwhile the generalized momenta along this congruence of the world lines are related\nto $S(\\xi)$ by\n \\begin{equation}\\label{gp16}\n p_i = \\frac{\\partial S}{\\partial \\xi^i} \\, ,\n \\end{equation}\nThe equations for the world lines obtain the form\n \\begin{equation}\\label{gp17}\n \\dot{\\xi}^i = g^{ijkl}\\frac{\\partial S}{\\partial\n \\xi^j}\\frac{\\partial S}{\\partial \\xi^k}\\frac{\\partial S}{\\partial\n \\xi^l} \\cdot \\lambda(\\xi) \\, ,\n \\end{equation}\nwere $\\lambda (\\xi )\\neq 0$.\n\nIn Physics the function $S(\\xi)$ is called \"action as a function of coordinates\"\\,\nand (\\ref{gp15}) is known as the Hamilton-Jacoby equation. In \\cite{10} the function\n$S(\\xi)$ was called the \\emph{World function}.\n\nIf there is a congruence of the world lines, then the evolution of every point in\nspace is known, particularly, the velocity field is known, but the energy\ncharacteristics of the material objects (observers) corresponding to a given world\nline are not known. The knowledge of the World function $S(\\xi)$ makes it possible\nto calculate the generalized momenta $p_i$, corresponding to the energy\ncharacteristics, and the invariant energy characteristic, $\\kappa(\\xi)$, which has\nalso the meaning of the local contraction-extension coefficient of the plane $H_4$\nspace.\n\nSo, if our world view is the classical mechanics, then any pair out of the three:\nWorld function, congruence of the world lines, Finsler geometry - gives us the\ncomplete knowledge of the World.\n\nLet us construct a twice contravariant tensor {\\it g}{\\it ij}{\\it (?)} in the\nfollowing way:\n \\begin{equation}\\label{gp18}\n g^{ij}(\\xi) = \\frac{1}{\\kappa(\\xi)^4} \\cdot g^{ijkl}\\frac{\\partial\n S}{\\partial \\xi^k}\\frac{\\partial S}{\\partial \\xi^l} \\, .\n \\end{equation}\nSince\n \\begin{equation}\\label{gp19}\n det(g^{ij}(\\xi))= - \\frac{4^4}{3^3 \\kappa(\\xi)^8} \\neq 0 \\, ,\n \\end{equation}\nthen everywhere where the geometry (\\ref{gp14}) is defined, one can construct a\ntensor $g_{ij}(\\xi)$ such that\n \\begin{equation}\\label{gp20}\n g^{ik}(\\xi)g_{kj}(\\xi)=\\delta^i_j \\, ,\n \\end{equation}\n \\begin{equation}\\label{gp21}\n g_{ij}(\\xi) = 4\\cdot \\left(\n \\begin{array}{cccc}\n -2\\left(\\frac{\\partial S}{\\partial \\xi^1}\\right)^2 & \\frac{\\partial S}{\\partial \\xi^1}\\frac{\\partial S}{\\partial \\xi^2} & \\frac{\\partial S}{\\partial \\xi^1}\\frac{\\partial S}{\\partial \\xi^3} & \\frac{\\partial S}{\\partial \\xi^1}\\frac{\\partial S}{\\partial \\xi^4}\n \\\\[9pt]\n \\frac{\\partial S}{\\partial \\xi^1}\\frac{\\partial S}{\\partial \\xi^2} & -2\\left(\\frac{\\partial S}{\\partial \\xi^2}\\right)^2 & \\frac{\\partial S}{\\partial \\xi^2}\\frac{\\partial S}{\\partial \\xi^3} & \\frac{\\partial S}{\\partial \\xi^2}\\frac{\\partial S}{\\partial \\xi^4} \\\\[9pt]\n \\frac{\\partial S}{\\partial \\xi^1}\\frac{\\partial S}{\\partial \\xi^3} & \\frac{\\partial S}{\\partial \\xi^2}\\frac{\\partial S}{\\partial \\xi^3} & -2\\left(\\frac{\\partial S}{\\partial \\xi^3}\\right)^2 & \\frac{\\partial S}{\\partial \\xi^3}\\frac{\\partial S}{\\partial \\xi^4} \\\\[9pt]\n \\frac{\\partial S}{\\partial \\xi^1}\\frac{\\partial S}{\\partial \\xi^4} & \\frac{\\partial S}{\\partial \\xi^2}\\frac{\\partial S}{\\partial \\xi^4} & \\frac{\\partial S}{\\partial \\xi^3}\\frac{\\partial S}{\\partial \\xi^4} & -2\\left(\\frac{\\partial S}{\\partial \\xi^4}\\right)^2\n \\end{array}\n \\right) .\n \\end{equation}\n\nNo doubt that in the same coordinate space $\\xi ^{1} ,\\xi ^{2} ,\\xi ^{3} ,\\xi ^{4}$\nsuch tensor $g_{ij}(\\xi)$ defines a Riemannian or pseudo Riemannian geometry with\nthe length element\n \\begin{equation}\\label{gp22}\n ds'' = \\sqrt{g_{ij}(\\xi)d\\xi^id\\xi^j} \\, .\n \\end{equation}\n\nThe construction of tensor $g_{ij}(\\xi)$ leads directly to the conclusion: the\nchange of geometry (\\ref{gp14}) to the geometry (\\ref{gp22}) does not lead to the\nchange of the initial congruence of the world lines and corresponding World function\n$S(\\xi)$.\n\nTherefore, in our concept one and the same World, i.e. the pair \\{World function;\ncongruence of the world lines\\}, corresponds to a whole class of related but\nqualitatively different Finsler geometries.\n\n\\section{Analyticity condition and the Minkowski space}\n\nLet the World function $S(\\xi)$ be the (unity) component of an analytical function\nof the $H_4$ variable in the orthogonal basis (\\ref{gp4}), that is\n \\begin{equation}\\label{gp23}\n S(\\xi) = \\frac{1}{4} \\left[ f^1(\\xi^1) + f^2(\\xi^2) + f^3(\\xi^3) + f^4(\\xi^4) \\right] \\, .\n \\end{equation}\nThen\n \\begin{equation}\\label{gp24}\n g^{ijkl}\\frac{\\partial S}{\\partial \\xi^i}\\frac{\\partial S}{\\partial\n \\xi^j}\\frac{\\partial S}{\\partial \\xi^k}\\frac{\\partial S}{\\partial\n \\xi^l} = \\frac{\\partial f^1(\\xi^1)}{\\partial \\xi^1}\\frac{\\partial\n f^2(\\xi^2)}{\\partial \\xi^2}\\frac{\\partial f^3(\\xi^3)}{\\partial\n \\xi^3}\\frac{\\partial f^4(\\xi^4)}{\\partial \\xi^4} = \\kappa(\\xi)^4 > 0 \\, ,\n \\end{equation}\nand this leads to the limitation on the functions, {\\it f}{\\it i}:\n \\begin{equation}\\label{gp25}\n \\frac{\\partial f^1(\\xi^1)}{\\partial \\xi^1}\\frac{\\partial\n f^2(\\xi^2)}{\\partial \\xi^2}\\frac{\\partial f^3(\\xi^3)}{\\partial\n \\xi^3}\\frac{\\partial f^4(\\xi^4)}{\\partial \\xi^4} > 0 \\, .\n \\end{equation}\n\nIt follows from (\\ref{gp24}) that the space with the length element (\\ref{gp14}) can\nbe obtained from the space with the length element (\\ref{gp6}) with the help of the\nconformal transformation, which means that the condition of the analyticity of the\nWorld function can be treated in a sense as the condition of the conformal symmetry.\n\nLet us construct tensor $g_{ij}(\\xi)$ following the algorithm developed in the\nprevious section. It turns out that in a region where functions $f^i$ have no\nsingularities there will always be such a coordinate system $x^0, \\, x^1, \\, x^2, \\,\nx^3$ in which the length element $ds''$ has a form\n \\begin{equation}\\label{gp26}\n ds'' = \\sqrt{(x^0)^2 - (x^1)^2 - (x^3)^2 - (x^3)^2} \\, .\n \\end{equation}\n\nLet us express the coordinates $x^0,\\, x^1,\\, x^2,\\, x^3$ in terms of the initial\ncoordinates $\\xi ^{1} ,\\xi ^{2} ,\\xi ^{3} ,\\xi ^{4} $:\n\\begin{equation}\\label{gp27}\n\\left.\n\\begin{array}{l}\n x^0 = \\displaystyle\\frac{\\;\\, 1 \\;}{4} \\left( f^1(\\xi^1) + f^2(\\xi^2) + f^3(\\xi^3) +\nf^4(\\xi^4) \\right) \\, ,\\\\[12pt]\n x^1 = \\displaystyle\\frac{\\sqrt{3}}{4}\\left( f^1(\\xi^1) + f^2(\\xi^2) - f^3(\\xi^3) -\nf^4(\\xi^4) \\right) \\, ,\\\\[12pt]\n x^2 = \\displaystyle\\frac{\\sqrt{3}}{4}\\left( f^1(\\xi^1) - f^2(\\xi^2) + f^3(\\xi^3) -\nf^4(\\xi^4) \\right) \\, ,\\\\[12pt]\n x^3 =\\displaystyle \\frac{\\sqrt{3}}{4}\\left( f^1(\\xi^1) - f^2(\\xi^2) - f^3(\\xi^3) +\nf^4(\\xi^4) \\right) \\, .\n\\end{array}\n\\right\\}\n\\end{equation}\n\nTherefore, to obtain the non-trivial curving of the space-time one should use the World functions with the broken conformal symmetry.\n\n\\section{Newtonian potential}\n\nLet us show that there are World functions that lead to the non-trivial pseudo\nRiemannian 4-dimensional spaces. Let us regard a function\n \\begin{equation}\\label{gp28}\n S(\\xi) = \\frac{1}{4}\\left( \\xi^1 + \\xi^2 + \\xi^3 + \\xi^4 \\right) +\n \\alpha\\cdot\\psi(\\varrho) \\, ,\n \\end{equation}\nwhere $\\alpha$ is the parameter characterizing the break of the analyticity of the\nWorld function (the break of the conformal symmetry in the $H_4$ space), $\\psi$ is\nan arbitrary function of a single argument\n \\begin{equation}\\label{gp29}\n \\varrho = \\sqrt{(y^1)^2+(y^2)^2+(y^3)^2} \\, ,\n \\end{equation}\nand $y^0,\\, y^1,\\, y^2,\\, y^3$ are the coordinates in the orthogonal basis $1, j, k,\njk$:\n \\begin{equation}\\label{gp30}\n \\left.\n \\begin{array}{c}\n y^0 = \\displaystyle \\frac{1}{4}(\\xi^1+\\xi^2+\\xi^3+\\xi^4) \\, , \\\\[12pt]\n y^1 = \\displaystyle \\frac{1}{4}(\\xi^1+\\xi^2-\\xi^3-\\xi^4) \\, , \\\\[12pt]\n y^2 = \\displaystyle \\frac{1}{4}(\\xi^1-\\xi^2+\\xi^3-\\xi^4) \\, , \\\\[12pt]\n y^3 = \\displaystyle \\frac{1}{4}(\\xi^1-\\xi^2-\\xi^3+\\xi^4) \\, .\n \\end{array}\n \\right\\} \\,\n \\end{equation}\nThen the derivatives of the World functions over the coordinates $\\xi^i$ can be\nexpressed in the following way:\n\\begin{equation}\\label{gp31}\n\\left.\n\\begin{array}{c}\n\\displaystyle\\frac{\\partial S}{\\partial \\xi^1} =\n\\displaystyle \\frac{1}{4}\\left[1+\\frac{\\alpha}{\\varrho}\\frac{d\\psi}{d\\varrho}\\left( y^1+y^2+y^3 \\right)\\right] \\, , \\\\[15pt]\n\\displaystyle\\frac{\\partial S}{\\partial \\xi^2} =\n\\displaystyle \\frac{1}{4}\\left[1+\\frac{\\alpha}{\\varrho}\\frac{d\\psi}{d\\varrho}\\left( y^1-y^2-y^3 \\right)\\right] \\, , \\\\[15pt]\n\\displaystyle\\frac{\\partial S}{\\partial \\xi^3} =\n\\displaystyle \\frac{1}{4}\\left[1+\\frac{\\alpha}{\\varrho}\\frac{d\\psi}{d\\varrho}\\left( -y^1+y^2-y^3 \\right)\\right] \\, , \\\\[15pt]\n\\displaystyle\\frac{\\partial S}{\\partial \\xi^4} =\n\\displaystyle\\frac{1}{4}\\left[1+\\frac{\\alpha}{\\varrho}\\frac{d\\psi}{d\\varrho}\\left(\n-y^1-y^2+y^3 \\right)\\right] \\, .\n\\end{array}\n\\right\\} \\,\n\\end{equation}\n\nLet us calculate the components of the metric tensor in coordinates $y^0,\\, y^1,\\,\ny^2,\\, y^3$ using the invariance of the square of the length element\n \\begin{equation}\\label{gp32}\n g_{ij}(\\xi) d\\xi^id\\xi^j = \\tilde{g}_{ij}(y)dy^idy^j\n \\end{equation}\nGrouping the terms, one gets\n \\begin{equation}\\label{gp33}\n \\tilde{g}_{00}= 1 - 3\\alpha^2\\left( \\frac{d\\psi}{d\\varrho} \\right)^2\n \\, , \\qquad \\tilde{g}_{\\beta\\beta_-}=-3\\left\\{1 + \\alpha^2\\left(\n \\frac{d\\psi}{d\\varrho} \\right)^2\\left[ 1 -\n \\frac{4(y^\\alpha)^2}{3\\rho^2}\\right]\\right\\} \\, ,\n \\end{equation}\n \\begin{equation}\\label{gp34}\n 2\\tilde{g_{0\\beta}} = - 4 \\left[\n \\alpha\\frac{d\\psi}{d\\varrho}\\frac{\\;\\; y^\\beta}{\\varrho} +\n 3\\alpha^2\\left(\\frac{d\\psi}{d\\varrho}\\right)^2\\cdot\\frac{y^1y^2y^3}{y^\\beta\\varrho^2}\n \\right] \\, ,\n \\end{equation}\n \\begin{equation}\\label{gp35}\n 2\\tilde{g}_{\\beta\\gamma} = - 4 \\left[ 3\n \\alpha\\frac{d\\psi}{d\\varrho}\\frac{\\;\\, y^\\delta}{\\varrho} +\n \\alpha^2\\left(\\frac{d\\psi}{d\\varrho}\\right)^2\\cdot\\frac{y^\\beta\n y^\\gamma}{\\varrho^2} \\right] \\, ,\n \\end{equation}\nwhere $\\beta,\\, \\gamma,\\, \\delta,\\, = 1, 2, 3$; $\\beta\\equiv\\beta_-$ but no\nsummation is performed here; in the last formula all the indices $\\beta,\\, \\gamma,\\,\n\\delta\\,$ are different.\n\n If $\\alpha = 0$, then\n \\begin{equation}\\label{gp36}\n (\\tilde{g}_{ij}) = diag(1,-3,-3,-3) \\, .\n \\end{equation}\nThis means that the real physical coordinates $x^0,\\, x^1,\\, x^2,\\, x^3$ of the\nspace-time are expressed by the coordinates $y^0,\\, y^1,\\, y^2,\\, y^3$ in the\nfollowing way\n \\begin{equation}\\label{gp37}\n x^0 = y^0 \\, , \\qquad x^\\beta = \\sqrt{3}\\cdot y^\\beta \\, .\n \\end{equation}\n\nLet us pass to the physical coordinates $x^0,\\, x^1,\\, x^2,\\, x^3$:\n \\begin{equation}\\label{gp38}\n \\tilde{g}_{ij}(y)dy^idy^j = \\bar{g}_{ij}(x)dx^idx^j \\, ,\n \\end{equation}\nwhere\n \\begin{equation}\\label{gp39}\n \\bar{g}_{00} = \\tilde{g}_{00} \\, , \\qquad \\bar{g}_{0\\beta} =\n \\frac{1}{\\sqrt{3}}\\cdot \\tilde{g}_{0\\beta} \\, , \\qquad \\bar{g}_{\\beta\\gamma} =\n \\frac{1}{3}\\cdot \\tilde{g}_{\\beta\\gamma} \\,.\n \\end{equation}\n Let us denote\n \\begin{equation}\\label{gp40}\n r = \\sqrt{(x^1)^2+(x^2)^2+(x^3)^2} \\equiv \\sqrt{3}\\cdot\\varrho \\, ,\n \\end{equation}\nThen\n \\begin{equation}\\label{gp41}\n \\bar{g}_{00}= 1 - 9\\alpha^2\\left( \\frac{d\\psi}{dr} \\right)^2 \\, ,\n \\qquad \\bar{g}_{\\beta\\beta_-}=-\\left\\{1 + 3\\alpha^2\\left(\n \\frac{d\\psi}{dr} \\right)^2\\left[ 1 -\n \\frac{4(x^\\alpha)^2}{3r^2}\\right]\\right\\} \\, ,\n \\end{equation}\n \\begin{equation}\\label{gp42}\n 2\\bar{g_{0\\beta}} = - 4 \\left[ \\alpha\\frac{d\\psi}{dr}\\frac{\\;\\;\n x^\\beta}{r} +\n 3\\sqrt{3}\\alpha^2\\left(\\frac{d\\psi}{dr}\\right)^2\\cdot\\frac{x^1x^2x^3}{x^\\beta\n r^2} \\right] \\, ,\n \\end{equation}\n \\begin{equation}\\label{gp43}\n 2\\bar{g}_{\\beta\\gamma} = - 4 \\left[ \\sqrt{3}\n \\alpha\\frac{d\\psi}{dr}\\frac{\\;\\, x^\\delta}{r} +\n \\alpha^2\\left(\\frac{d\\psi}{dr}\\right)^2\\cdot\\frac{x^\\beta x^\\gamma}{r^2} \\right] \\, .\n \\end{equation}\n\nThe metric tensor $\\bar{g}_{ij} (x)=\\bar{g}_{ij} (x^{1} ,x^{2} ,x^{3} )$ depends\nonly on the space coordinates $x^1,x^2,x^3$, and this corresponds to the stationary\ngravitational field, stationary Universe. The probe particle of mass $m$ moves along\nthe geodesic of the pseudo Riemannian space with metric tensor $\\bar{g}_{ij} (x^{1}\n,x^{2} ,x^{3} )$.\n\nLet a particle move in a fixed frame and have velocity much less than the light\nvelocity, $c$:\n \\begin{equation}\\label{gp44}\n \\frac{dx^\\beta}{dt} = v^\\beta \\, , \\qquad |v^\\beta| \\ll c \\, ,\n \\end{equation}\nThe gravitational fields are weak, that is the condition $|v^{\\beta } |<<1$ remains\nvalid for all the time of the particle motion. Let us obtain the Lagrange function,\n{\\it L}, to describe such non-relativistic motion of the probe particle in the weak\ngravity field. To do this, develop the right hand side of the expression\n \\begin{equation}\\label{gp45}\n L = -mc\\cdot \\frac{\\sqrt{\\bar{g}_{ij}(x^1,x^2,x^3)dx^idx^j}}{dt}\n \\end{equation}\nWithin the accuracy of $\\left(\\frac{v}{c}\\right)^2$\n\\begin{equation}\\label{gp46}\nL = -mc^2 \\sqrt{\\bar{g}_{00}} \\cdot \\sqrt{1+ \\frac{1}{\\bar{g}_{00}}\\left(\n2\\bar{g}_{0\\beta}\\frac{v^\\beta}{c} + \\bar{g}_{\\beta\\gamma}\\frac{v^\\beta\nv^\\gamma}{c^2} \\right)} \\, ,\n\\end{equation}\n\\begin{equation}\\label{gp47}\nL \\simeq -mc^2 \\sqrt{\\bar{g}_{00}} \\cdot \\left\\{1+ \\frac{1}{2\\bar{g}_{00}}\\left(\n2\\bar{g}_{0\\beta}\\frac{v^\\beta}{c} + \\bar{g}_{\\beta\\gamma}\\frac{v^\\beta\nv^\\gamma}{c^2} \\right) - \\frac{1}{8\\bar{g}^2_{00}}\\left(\n2\\bar{g}_{0\\beta}\\frac{v^\\beta}{c} \\right)^2\\right\\} \\, .\n\\end{equation}\nOpening the brackets in the right hand side, we get an additive term which is the\nfull time derivative of a certain function {\\it f(r)}, it depends linearly on the\nvelocity components and, thus, it can be omitted. Leaving the same designation for\nthe Lagrange function, we get\n\\begin{equation}\\label{gp48}\nL \\simeq -mc^2 \\sqrt{\\bar{g}_{00}} \\cdot \\left\\{1+ \\frac{1}{2\\bar{g}_{00}}\\cdot\n\\bar{g}_{\\beta\\gamma}\\frac{v^\\beta v^\\gamma}{c^2} - \\frac{1}{8\\bar{g}^2_{00}}\\left(\n2\\bar{g}_{0\\beta}\\frac{v^\\beta}{c} \\right)^2\\right\\} \\, .\n\\end{equation}\n\nOur goal is the Lagrange function of the form\n \\begin{equation}\\label{gp49}\n L = \\frac{m\\vec{v}^2}{2} - U(\\vec{x}) \\, ,\n \\end{equation}\nwhere $U(\\vec{x})$ is the potential energy of the probe particle, $\\vec{x}\\equiv\n(x^{1} ,x^{2} ,x^{3} ),~ \\vec{v}\\equiv (v^{1} ,v^{2} ,v^{3} ),~ r^{2} =\n\\vec{x}^{\\,2}$, $\\vec{v}^{\\,2} =(v^{1} )^{2} +(v^{2} )^{2} +(v^{3} )^{2} \\equiv\nv^{2} $. To reach it we have to make some assumptions about the correlation between\nthe parameter, $\\alpha$ and light velocity:\n \\begin{equation}\\label{gp50}\n \\alpha = \\frac{\\nu}{c} \\, , \\quad \\hbox{when} \\quad c\\rightarrow\n \\infty \\quad \\alpha\\rightarrow 0 \\, .\n \\end{equation}\nBesides, let $\\alpha$ be of the same order (or smaller) with the relation\n$\\left|\\displaystyle\\frac{v}{c}\\right|$. Then leaving only the terms that don't\ndisappear at $c\\to \\infty $ in the (\\ref{gp48}), one gets\n \\begin{equation}\\label{gp51}\n L \\simeq -mc^2 + mc^2 \\frac{9}{2}\\frac{\\nu^2}{c^2} \\left(\n \\frac{d\\psi}{dr} \\right)^2 + m\\cdot \\frac{v^1v^1+v^2v^2+v^3v^3}{2}\\, .\n \\end{equation}\nSince $(-mc^2)$ is a full time derivative of function $(-mc^2\\cdot t)$, we omit it\nand get\n \\begin{equation}\\label{gp52}\n L \\simeq \\frac{m\\vec{v}^2}{2} + \\frac{9m\\nu^2}{2} \\left(\n \\frac{d\\psi}{dr} \\right)^2 \\, .\n \\end{equation}\n\nLet a mass $M$ be motionless in the frame origin, and then the potential energy of\nthe probe particle with mass $m$ located at $x^1,x^2,x^3$ is equal to\n \\begin{equation}\\label{gp53}\n U(r) = - \\gamma \\frac{mM}{r} \\, ,\n \\end{equation}\nwhere $\\gamma$ is the gravitational constant. Comparing (\\ref{gp49}) and\n(\\ref{gp52}), we get the equation for $\\psi(r)$:\n\\begin{equation}\\label{gp54}\n\\frac{9m\\nu^2}{2} \\left(\\frac{d\\psi}{dr} \\right)^2 = \\gamma \\frac{mM}{r} \\qquad\n\\Rightarrow \\qquad \\frac{d\\psi}{dr} = \\pm \\frac{\\sqrt{2\\gamma\nM}}{3\\nu}\\frac{1}{r^{1\/2}} \\, .\n\\end{equation}\nTherefore,\n\\begin{equation}\\label{gp55}\n\\psi(r) = \\pm \\frac{2\\sqrt{2\\gamma M}}{3\\nu}\\cdot r^{1\/2} + \\psi_0 \\qq (\\psi_0 =\nconst).\n\\end{equation}\n\nFinally, the World function is equal to\n\\begin{equation}\\label{gp56}\nS = x^0 \\pm \\frac{2\\sqrt{2\\gamma M}}{3c}\\cdot r^{1\/2} + C_0 \\qq (C_0 = const),\n\\end{equation}\nWhen it performs a conformal transformation of the length element of the plane\nBerwald-Moor space, it induces a pseudo Riemannian geometry in the Minkowski space.\nFor a non-relativistic probe particle of mass {\\it m}, this geometry gives the\nmotion equations for the Kepler problem for the point mass {\\it M} located in the\norigin of the space frame.\n\n The more complicated World function, maybe also leading to the stationary Universe, has the form\n\\begin{equation}\\label{gp57}\nS(\\xi) = \\frac{1}{4}\\left( \\xi^1 + \\xi^2 + \\xi^3 + \\xi^4 \\right)\\left[\n1+\\alpha_1\\cdot\\psi_1(\\varrho) \\right] + \\alpha_2\\cdot\\psi_2(\\varrho) \\, ,\n\\end{equation}\nwhere $\\alpha_A$ are the parameters of the analyticity break of the World function\n(parameters of the conformal symmetry break in the $H_4$ space), $\\psi_A$ are the\narbitrary functions of single argument $\\varrho$ (\\ref{gp29}), (\\ref{gp30}).\n\n \\section*{Conclusion}\n\nThe results obtained in this paper point at the deep correlation between the\nEinstein geometries and Finsler spaces with Berwald-Moor metric. We managed to find\nthe concrete Finsler space with the Berwald-Moor metric which in the limit appeared\nto be related to the curved pseudo Riemannian space with the Newtonian gravitational\npotential. This fact points at the principal possibility to built more interesting\nconstructions, particularly, such Finsler spaces whose limit cases would be the\nknown relativistic solutions.\n\n\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nI was asked to talk about thermal radiation from\nisolated neutron stars (NSs). In this meeting, George Pavlov\nreviewed the X-ray properties of pulsars and thermally emitting NSs\n(see Kaplan et al.~2011), and Wynn Ho discussed central compact\nobjects and their magnetic fields (see Halpern \\& Gotthelf 2010;\nShabaltas \\& Lai 2012; Vigano \\& Pons 2012). Recent works on\ntheoretical modelling of NS surface emission can be found in Potekhin\net al.~(2012) (see also Pavlov et al.~1995; Harding \\& Lai 2006 and\nvan Adelsberg \\& Lai 2006 for reviews). Since these subjects were\nadequately covered in the meeting, I decided to focus on a different\ntopic that did not receive much attention in this meeting but is\nlikely to become increasingly important in the coming decade.\n\n\nMerging NS binaries have been studied since 1970s, with major\nactivities in the relativity community since the early 1990s because\nof their importance as a source of gravitational waves (GWs)\n(e.g. Cutler et al.~1993). They are of great current interest for two\nreasons: (i) Merging NS\/NS or NS\/Black-Hole (BH) binaries have been\nidentified as the leading candidate for the central engine of short\nGRBs\n(Berger 2011). They are also expected to produce optical and radio\ntransients that may be detected by wide-field, fast imaging telescopes\nthat are coming online (e.g. PTF, LSST) in the next few years\n(Nissanke et al.~2012). (ii) After several decades of developmemt and\npromise,\ngravitational wave astronomy in the Hz-kHz band may finally take off\nin the next decade. The initial LIGO reached the design sensitivity\n($h_c\\simeq 10^{-21}$) in 2006, and the enhanced LIGO (with a factor of 2\nreduction in $h_c$) is taking or analysing data. The Advanced LIGO and\nVIRGO are expected to begin observations in 2015 and reach full\nsensitivity (a factor of 10 reduction in $h_c$) in 2018-19 --- at which\ntime the detection of GWs from many merging NS binaries seems\nguaranteed.\n\nThe last three minutes of a NS binary's life may be divided into two\nphases: the inspiral phase, producing quasi-periodic GWs, and the\ncoalescence phase, where physical collision results in ``messy'' GWs.\nThe recent years, 3D simulations of the final merger in full general\nrelativity (GR) have become possible (see Shibata \\& Taniguchi 2006;\nFoucart et at.~2012; Sekiguchi et al.~2012). It has long been\nrecognized that the final merger waveforms can provide a useful probe of NS\nequation of state (EOS; e.g., Cutler et al.~1993; Bildsten \\& Cutler\n1992; Lai \\& Wiseman 1996; Wiggins \\& Lai 2000). The idea is \nsimple: By measuring the ``cut-off'' frequency $\\propto\n(GM_t\/R^3)^{1\/2}$ associated with binary contact or tidal disruption,\ncombined with the precise mass measurement from the inspiral waveform,\none can obtain the NS radius (see Bauswein et al.~2012 for recent \nsimulations which put such a idea into concrete footing; see\nalso Sekiguchi et al.~2012; Faber \\& Rasio 2012 for reviews). \n\nIn the following sections I will focus on the pre-merger phase.\n\n\n\\section{Hydrodynamics of merging NS binaries}\n\nPrior to binary merger, tidal effects may affect the orbital decay and\nthe GWs. There are two types of tides: {\\it equilibrium tides} and\n{\\it dynamical tides}. The equilibrium tides correspond to global deformation \nof the NS, which leads to the interaction potential between the two stars\n(with the NS mass $M$ and radius $R$, the companion mass $M'$ -- treated as a \npoint mass, and the binary separation $a$) \n\\begin{equation}\nV(r)=-{MM'\/a}-{\\cal O}\\left(k_2{{M'}^2R^5\/a^6}\\right),\n\\end{equation}\nwhere $k_2$ is the so-called\nLove number. This would lead to a correction to the number of GW cycles,\n$dN=dN^{(0)}\\left[1-{\\cal O}(k_2M'R^5\/Ma^5)\\right]$. For a Newtonian\npolytropic NS model, simple analytic expressions can be found in Lai et\nal.~(1994). Recent semi-analytic GR calculations of such equilibrium\ntidal effects (including the more precise determination of the Love number)\ncan be found in numerous papers (e.g., Flanagan \\& Hinderer 2008;\nBinnington \\& Poisson 2009; Damour \\& Nagar 2009; Penner et al.~2012, \nFerrari et al.~2012). Obviously\nthis effect is only important at small orbital separations (just prior\nto merger) -- there is some prospect of measuring this, thereby\nconstraining the EOS, but it may be challenging (Damour et al.~2012).\nAt small orbital separations, the quadrupole approximation is not valid; \ntherefore one\nmust use the numerically computed GR quasi-equilibrium binary sequences\nto characterize the tidal effect -- such sequences have been\nconstructed by several groups since the 1990s (e.g., Baumgarte et al.~1998;\nUryu et al.~2009) or use fully numerical simulations.\n\nAnother aspect of the equilibrium tide concerns tidal dissipation,\nwhich leads to a lag of the tidal bulge with respect to the binary\naxis. It was shown already in the 1990s (Bildsten \\& Cutler 1992;\nKochanek 1992) that because of the rapid GW-driven orbital decay,\nviscous tidal lag cannot synchronize the NS spin. Thus the NS will\nbe close to irrotational (approximated as a Riemann-S ellipsoid;\nLai et al.~1994; Wiggins \\& Lai 2000; Ferrari et al.~2012). \nNear the final phase\nof the inspiral, the rapid orbital decay gives rise to a finite lag\nangle (even with zero viscosity), but this cannot synchronize the NS\n(Lai \\& Shapiro 1995; Dall'Osso \\& Rossi 2012).\n\n\nThe situation is more complicated for {\\bf dynamical tides}, which\nmanifest as resonant excitations of internal oscillations of the NS: As\ntwo NSs spiral in, the orbit can momentarily come into resonance with\nthe normal modes (frequency $\\omega_\\alpha$) of the NS:\n\\begin{equation}\n\\omega_\\alpha=m\\Omega_{\\rm orb},\\qquad m=2,3,\\cdots\n\\end{equation}\nBy drawing energy from the orbital\nmotion and resonantly exciting the modes, the rate of inspiral is\nmodified, giving rise to a phase shift in the gravitational\nwaveform. This problem was studied by Reisenegger \\& Goldreich (1994),\nLai (1994) and Shibata (1994) in the case of non-rotating NSs, where\nthe only modes that can be resonantly excited are g-modes (with\ntypical mode frequencies$\\lo 100$~Hz). It was found that the effect is\nsmall for typical NS parameters (mass $M=1.4M_\\odot$ and radius\n$R=10$~km) because the coupling between the g-mode and the tidal\npotential is weak. Ho \\& Lai (1999) studied the effect of NS rotation,\nand found that the g-mode resonance can be strongly enhanced even by a\nmodest rotation (e.g., the phase shift in the waveform $\\Delta\\Phi$\nreaches up to 0.1~radian for a spin frequency $\\nu_s\\lo 100$~Hz).\nThey also found that for a rapidly rotating NS ($\\nu_s\\go 500$~Hz),\nf-mode resonance becomes possible (since the inertial-frame f-mode\nfrequency can be significantly reduced by rotation) and produces a\nlarge phase shift. In addition, NS rotation gives rise to r-mode\nresonance whose effect is appreciable only for very rapid (near\nbreakup) rotations. Lai \\& Wu (2006) further studied resonant\nexcitations of other inertial modes (of which r-mode is a member) and\nfound similar effects. Flanagan \\& Racine (2006) studied the\ngravitomagnetic resonant excitation of r-modes and and found that the\npost-Newtonian effect is more important than the Newtonian tidal\neffect (and that the phase shift reaches 0.1~radian for $\\nu_s\\sim\n100$~Hz). Tsang et al.~(2012) examined crustal modes and found that \nthe GW phase correction is small\/modest and suggested that tidal resonance\ncould shatter the NS crust, giving rise to the pre-cursor of short GRBs.\nTaken together, these studies suggest that for canonical NS\nparameters ($R \\simeq 10$~km, $\\nu_s\\lo 100$~Hz), tidal resonances\nhave a small effect on the gravitational waveform during binary\ninspiral. However, it is important to keep in mind that the effect is \na strong function of $R$ (e.g., $\\Delta\\Phi\\propto R^4$ for g-modes\nand $\\propto R^{3.5}$ for inertial modes). A larger radius ($R\\simeq \n15$~km), appropriate for stiff EOS, \nwould make the effect important. In the case of g-modes, the magnitude\nof the effect depends on the symmetry energy of nuclear matter \nand could be non-negligible (W. Newton \\& D. Lai 2013, in prep).\n\n\n\\section{Electrodynamics of merging NS binaries}\n\nFor magnetic NSs, magnetic interactions may play a role. If the\nbinary is embedded in a vacuum, then the interaction potential is\n$V(r)=-MM'\/a-{\\cal O}(\\mu\\mu'\/a^3)$ (where $\\mu,\\mu'$ are the magnetic\ndipole moments of the two stars). It is easy to check that such\nmagnetic interaction would lead to negligible effect on the GWs unless\nboth NSs have superstrong fields ($\\gg 10^{15}$~G) -- this is unlikely\n(e.g., the double pulsars PSR J0737-3039 has $10^{10}$~G for pulsar A\nand $2\\times 10^{13}$ for pulsar B). \n\n\\begin{figure}[b]\n\\vspace*{-0.8 cm} \n\\begin{center}\n \\includegraphics[width=3.4in]{f1.eps}\n\\vspace*{-1.0 cm} \n \\caption{DC circuit (unipolar induction) \nmodel of magnetic interactions in binary \nsystems {\\it a la} Goldreich \\& Lynden-Bell (1969).}\n \\label{fig1}\n\\end{center}\n\\end{figure}\n\n\nOf course, as in the case of isolated pulsars,\nthe circumbinary environment cannot be vacuum.\nThe following discussion is based on Lai (2012).\nConsider a binary system consisting of a magnetic NS (the\n``primary'', with mass $M$, radius $R$, spin $\\Omega_s$, and magnetic\ndipole moment $\\mu$) and a non-magnetic companion (mass $M_c$, radius\n$R_c$). The orbital angular\nfrequency is $\\Omega$. The magnetic field strength at the surface of\nthe primary is $B_\\star=\\mu\/R^3$. The whole binary system is embedded\nin a tenuous plasma (magnetosphere). We assume for simplicity that\n${\\bf\\Omega}$, ${\\bf\\Omega_s}$ and ${\\mbox{\\boldmath $\\mu$}}$ are all aligned.\nThe motion of the non-magnetic companion relative to the magnetic field of the\nprimary produces an EMF ${\\cal E} \\simeq 2R_c |E|$, where ${\\bf E}= \n{\\bf v}_{\\rm rel}\\times {\\bf B}\/c$, with\n${\\bf v}_{\\rm rel}=(\\Omega-\\Omega_s)a\\,{\\hat{\\mbox{\\boldmath $\\phi$}}}$ and\n${\\bf B}=(-\\mu\/a^3){\\hat{\\bf z}}$.\nThis gives\n${\\cal E}\\simeq {(2\\mu R_c\/ca^2)}\\Delta\\Omega$,\nwhere $\\Delta\\Omega=\\Omega-\\Omega_s$\\footnote{If the magnetic field does\nhave time to fully penetrate the companion star due to the rapid orbital decay,\nthen the EMF will be reduced by a factor of order the ratio of the skin depth\nand the stellar radius. I thank Anatoly Spitkovsky for\npointing this out to me at the IAU conference.}. \nThe EMF drives a current along the magnetic\nfield lines in the magnetosphere, connecting the primary and the companion\nthrough two flux tubes. The current in the circuit is given by\n${\\cal I}={{\\cal E}\/({\\cal R}_{\\rm tot})}$,\nwhere the total resistance of the circuit is\n${\\cal R}_{\\rm tot}={\\cal R}+{\\cal R}_c+2{\\cal R}_{\\rm mag}$,\nwith ${\\cal R},\\,{\\cal R}_c,\\,{\\cal R}_{\\rm mag}$ the resistances of the magnetic\nstar, the companion and the magnetosphere, respectively.\nThese resistances depend on the properties of the binary components and the\nmagnetosphere, and can vary widely for different types of systems.\nThe energy dissipation rate of the system is then\n$\\dot E_{\\rm diss}=2{\\cal I}^2R_{\\rm tot}={2{\\cal E}^2\/{\\cal R}_{\\rm tot}}$,\nwhere the factor of 2 accounts for both the upper and lower sides of the circuit.\n\nThe total magnetic force (in the azimuthal direction) on the companion is\n$F_\\phi\\simeq (2R_c) (2{\\cal I}B_z\/c)$, with $B_z=-\\mu\/a^3$.\nThus the torque acting on the binary's orbital angular momentum is\n$T=\\dot J_{\\rm orb}\\simeq ({4\/c})a\\,R_c{\\cal I}B_z\\simeq\n-({4\\mu R_c\/ca^2})({{\\cal E}\/{\\cal R}_{\\rm tot}})$.\nThe torque on the primary's spin is $I\\dot\\Omega_s=-T$ (where $I$ is\nthe moment of inertia).\nThe orbital energy loss rate associated with $T$ is then\n$\\dot E_{\\rm orb}=T\\Omega$.\n\nThe equations above show that the binary\ninteraction torque and energy dissipation associated with the DC\ncircuit increase with decreasing total resistance ${\\cal R}_{\\rm tot}$.\nIs there a problem for the DC model when ${\\cal R}_{\\rm tot}$ is too small?\nThe answer is yes.\nThe current in the circuit produces a toroidal magnetic field, which\nhas the same magnitude but opposite direction above and below\nthe equatorial plane. The toroidal field just above the\ncompanion star (in the upper flux tube) is $B_{\\phi+}\\simeq -(2\\pi\/c){\\cal J}_r$,\nwhere ${\\cal J}_r\\simeq -4{\\cal I}\/(\\pi R_c)$ is the (height-integrated) surface current.\nThus the azimuthal twist of the flux tube is\n$\\zeta_\\phi=-{B_{\\phi+}\/B_z}=\n={16 v_{\\rm rel}\/(c^2{\\cal R}_{\\rm tot})}$,\nwhere $v_{\\rm rel}=a\\Delta\\Omega=a(\\Omega-\\Omega_s)$.\nClearly, when ${\\cal R}_{\\rm tot}$ is less than $16v_{\\rm rel}\/c^2$,\nthe flux tube will be highly twisted.\n\nGoldreich \\& Lynden-Bell (1969) speculated that the DC circuit would break\ndown when the twist is too large. (For the Jupiter-Io system\nparameters, the twist $|\\zeta_\\phi|\\ll 1$.) \nNumerous works have since confirmed that this is indeed the case.\nTheoretical studies and numerical simulations, usually carried out\nin the contexts of solar flares\nand accretion disks, have shown that as a flux tube is twisted beyond\n$\\zeta_\\phi\\go 1$, the magnetic pressure associated with $B_\\phi$\nmakes the flux tube expand outward and the magnetic fields open up,\nallowing the system to reach a lower energy state (e.g., Aly 1985; \nLynden-Bell \\& Boily 1994;\nLovelace et al.~1995; Uzdensky et al.~2002).\nThus, a DC circuit with $\\zeta_\\phi\\go 1$ cannot be realized: The\nflux tube will break up, disconnecting the linkage between the two\nbinary components.\nA binary system with ${\\cal R}_{\\rm tot}\\lo 16v_{\\rm rel}\/c^2$\ncannot establish a steady-state DC circuit.\nThe electrodynamics is likely rather complex, only\na quasi-cyclic circuit may be possible (Lai 2012; see Aly \\& Kuijpers 1990):\n(a) The magnetic field from the primary penetrates\npart of the companion, establishing magnetic linkage between the two\nstars; (b) The linked fields are twisted by differential rotation, generating\ntoroidal field from the linked poloidal field; (c) As the toroidal magnetic field\nbecomes comparable to the poloidal field, the fields inflate and\nthe flux tube breaks, disrupting the magnetic linkage;\n(d) Reconnection between the inflated field lines relaxes the shear and restore\nthe linkage. The whole cycle repeats.\n\nIn general, we can use the dimensionless azimuthal twist $\\zeta_\\phi$\nto parameterize the magnetic torque and energy dissipation rate:\n\\begin{equation}\nT= {1\\over 2}aR_c^2B_zB_{\\phi+}\n= -\\zeta_\\phi{\\mu^2 R_c^2\\over 2a^5},\\quad\n\\dot E_{\\rm diss} = -T \\Delta\\Omega\n=\\zeta_\\phi\\Delta\\Omega {\\mu^2 R_c^2\\over 2a^5}.\n\\label{eq:emax}\\end{equation}\nThe maximum torque and dissipation are obtained by setting $\\zeta_\\phi\\sim 1$.\nIf the quasi-cyclic circuit discussed in the last paragraph is\nestablished, we would expect $\\zeta_\\phi$ to vary between $0$ and $\\sim 1$.\nNote that in the above, $T$ is negative since we\nare assuming $\\Omega>\\Omega_s$. \n\nGravitational wave (GW) emission drives the orbital decay of the \nNS binary, with timescale\n$t_{\\rm GW}={a\/|\\dot a|}=\n0.012\\left({a\/30\\,{\\rm km}}\\right)^{4}{\\rm s}$,\nwhere we have adopted $M=1.4M_\\odot$ and mass ratio\n$q=M_c\/M=1$.\nThe magnetic torque tends to spin up the primary when $\\Omega$$>$$\\Omega_s$.\nSpin-orbit synchronization is possible only if\nthe synchronization time $t_{\\rm syn}=I\\Omega\/|T|$ is less than\n$t_{\\rm GW}$ at some orbital radii. With\n$I=\\kappa M R^2$, we find\n\\begin{equation}\nt_{\\rm syn}={2\\kappa(1+q)\\over\\zeta_\\phi\\Omega}\n\\left(\\!{GM^2\\over B_\\star^2R^4}\\!\\right)\n\\!\\left(\\!{a\\over R_c}\\!\\right)^2\n\\simeq 2\\times 10^7\\zeta_\\phi^{-1}\\!\\left(\\!{B_\\star\\over 10^{13}\\,{\\rm G}}\n\\!\\right)^{\\!-2}\\!\\left({a\\over 30\\,{\\rm km}}\\right)^{7\/2}{\\rm s},\n\\label{eq:tsyn}\\end{equation}\nwhere on the right we have adopted $\\kappa=0.4$ and $R=R_c=10$~km.\nClearly, even with magnetar-like field strength ($B_\\star\\sim 10^{15}$~G) and\nmaximum efficiency ($\\zeta_\\phi\\sim 1$), spin-orbit synchronization cannot be\nachieved by magnetic torque. For the same reason, the effect of magnetic torque on the\nnumber of GW cycles during binary inspiral is small. \n\nThe energy dissipation rate is\n\\begin{equation}\n\\dot E_{\\rm diss}=\\zeta_\\phi\\left(\\!{v_{\\rm rel}\\over c}\\!\\right){B_\\star^2R^6\nR_c^2c\\over 2a^6}\n= 7.4\\times 10^{44}\\zeta_\\phi\\left(\\!{B_\\star\\over\n10^{13}\\,{\\rm G}}\\!\\right)^{\\!2}\\!\\left(\\!{a\\over 30\\,{\\rm km}}\\!\\right)^{\\!\\!-13\/2}\n\\!{\\rm erg\\,s}^{-1},\n\\end{equation}\nwhere on the right we have used $v_{\\rm rel}\\simeq a\\Omega$ (for\n$\\Omega_s\\ll \\Omega$) and adopted canonical parameters\n($M=M_c=1.4M_\\odot$, $R=R_c=10$~km).\nThe total energy dissipation per $\\ln a$ is\n\\begin{equation}\n{dE_{\\rm diss}\\over d\\ln a}=\\dot E_{\\rm diss}t_{\\rm GW}\n\\simeq 8.9\\times 10^{42}\\zeta_\\phi\\!\\left(\\!{B_\\star\\over\n10^{13}\\,{\\rm G}}\\!\\right)^{\\!2}\\!\\left({a\\over 30\\,{\\rm km}}\\right)^{\\!\\!-5\/2}\n\\!{\\rm erg}.\n\\end{equation}\nSome fraction of this dissipation will emerge as electromagnetic\nradiation counterpart of binary inspiral. It is possible that \nthis radiation is detectable at extragalactic distance. But this will depend \non the microphysics in the magnetosphere, including particle acceleration and \nradiation mechanism (e.g., Vietri 1996; Hansen \\& Lyutikov 2001).\n\nIf one assumes that the magnetosphere resistance is given by the\nimpedance of free space, ${\\cal R}_{\\rm mag}=4\\pi\/c$, then the corresponding twist\nis $\\zeta_\\phi=2v_{\\rm rel}\/(\\pi c)$, which satisfies our upper limit.\nWe then have\n\\begin{equation}\n\\dot E_{\\rm diss}=\\left(\\!{v_{\\rm rel}\\over c}\\!\\right)^2\\!\n{B_\\star^2R^6 R_c^2c\\over \\pi a^6}\n= 1.7\\times 10^{44}\\left(\\!{B_\\star\\over 10^{13}\\,{\\rm G}}\\!\\right)^{\\!2}\n\\!\\left({a\\over 30\\,{\\rm km}}\\right)^{\\!\\!-7}{\\rm erg\/s}.\n\\end{equation}\nThis is in agreement with the estimate of Lyutikov (2011).\n\nThe situation is similar for NS\/BH binaries.\nIn the membrane paradigm (Thorne et al.~1986), a BH\nof mass $M_H$ resembles a sphere of radius $R_c=R_H=2GM_H\/c^2$\n(neglecting BH spin)\nand impedance ${\\cal R}_H=4\\pi\/c$. Neglecting the resistances of the\nmagnetosphere and the NS, the azimuthal twist of the flux tube in the DC\ncircuit is\n$\\zeta_\\phi={4v_{\\rm rel}\/(\\pi c)}$,\nwhich satisfies our upper limit.\nThe energy dissipation rate is (cf. Lyutikov 2011; McWilliams \\& Levin 2011)\n\\begin{equation}\n\\dot E_{\\rm diss}=\\left(\\!{v_{\\rm rel}\\over c}\\!\\right)^2\\!\n{2B_\\star^2R^6 R_H^2c\\over \\pi a^6}\n\\simeq 5.7\\!\\times\\! 10^{42}\\!\\left(\\!{B_\\star\\over 10^{13}\\,{\\rm G}}\\!\\right)^{\\!2}\n\\!\\!\\left(\\!{M_H\\over 10M_\\odot}\\!\\right)^{\\!\\!\\!-4}\n\\!\\!\\!\\left(\\!{a\\over 3R_H}\\!\\right)^{\\!\\!-7}\\!\\!\\!{\\rm erg\\,s}^{-1},\n\\end{equation}\nwhere we have assumed $M_{BH}\/M\\gg 1$.\nAgain, it is uncertain whether this radiation is detectable for binaries\nat extragalactic distances.\n\n\\vspace{1ex}\n{\\bf Acknowledgements}:\nThis work has been supported in part by NSF grants AST-1008245 and\nAST-1211061, and NASA grants NNX12AF85G and NNX10AP19G.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nStrongly interacting systems in the high energy (or small $x$) limit\nare very nonlinear systems in spite of the smallness of\nthe coupling constant $\\alpha_{\\mathrm{s}}$. This is due to the large phase\nspace available for semihard gluon radiation that increases \nthe occupation numbers of gluonic modes in the hadron or nucleus \nwavefunction. Thus high energy scattering has to be understood in\nterms of gluon recombination and saturation that enforce the unitarity\nrequirements of the $S$-matrix. This happens naturally in\nthe Color Glass Condensate (CGC) effective theory of the\nhigh energy wavefunction. In the context of \ndeep inelastic scattering (DIS) the CGC \nleads to the dipole picture that naturally\ngives a consistent description of both inclusive\nand diffractive scattering.\nThe nonlinearities in high energy scattering are enhanced when the target\nis changed from a proton to a heavy nucleus. Thus there is a great\nopportunity to understand them by studying nuclear DIS in \nnew collider experiments, such as the EIC~\\cite{Deshpande:2005wd} \nor the LHeC~\\cite{Dainton:2006wd}. The particular process we discuss in this\npaper is diffractive DIS on nuclei.\n\n\n\nIn the Good-Walker~\\cite{Good:1960ba} picture of diffraction one needs to identify \nthe states\nthat diagonalize the imaginary part of the $T$-matrix. In the case of \nnuclear DIS at high energy these states are the ones with the virtual photon \nfluctuating into a dipole of a fixed size $r$ and with the nucleons in\nthe nucleus at fixed transverse positions $b_i$. In coherent diffraction\nthe nucleus is required to stay intact, which corresponds to performing\n the average over the nuclear wavefunction at the level of the\nscattering amplitude. \nAveraging the cross section, instead of the amplitude, \nover the nucleon positions allows for the nucleus\nto break up, giving the sum of incoherent and coherent cross sections,\ni.e. the quasielastic cross section. \nFor a more formal discussion of this we point the reader e.g. to \nRef.~\\cite{Caldwell:2009ke}.\nThe $t$-dependence of the incoherent cross section therefore directly \nprobes the fluctuations and correlations in the nuclear wavefunction, which\nhave turned out to be a crucial ingredient in understanding the initial \nconditions in heavy ion collisions \\cite{Miller:2007ri,*Alver:2010gr}.\n\nThe average gluon density probed in the \ncoherent process is very smooth, meaning that the cross section is dominated\nby small values of momentum transfer to the nucleus, $t \\sim - 1\/R_A^2$.\nMeasuring such a small momentum transfer accurately is \nvery challenging.\n At momentum scales corresponding to the nucleon \nsize $t \\sim - 1\/R_p^2$ the diffractive cross section is almost purely incoherent.\nThe larger momentum transfer should\nalso be easier to reconstruct experimentally even without measuring\nthe transverse momentum of the \nnuclear remnants, by accurately reconstructing the outgoing electron\nand $J\/\\Psi$ momenta and using momentum conservation.\nBy taking these processes into account in the detector design\none should be capable of measuring diffractive events at a higher accuracy\nthan was done at HERA.\nIn the dilute limit (for small dipoles) there is no multiple scattering, and the\nincoherent cross section is given by $A$ times the corresponding one for\nprotons. The\ndeviation of the $t$-slope from the proton measures the transverse size of the \nfluctuating areas in the nucleus. \n\n\nIn the black disc limit \nthe nucleus is smooth not only on average, but event-by-event, \nleading to a strong suppression of the incoherent cross section. Incoherent\ndiffraction gets contributions from the edge of the nucleus, making the\ncross section asymptotically behave as $\\sim A^{1\/3}$ in contrast to\n$\\sim A$ in the dilute limit.\nThe suppression in the\nnormalization relative to the proton is a measure of the approach to the unitarity\nlimit in the dipole cross section. It is a clear signal of how individual nucleons\nhave lost their identity in the sense that they cannot be resolved by the virtual photon.\nIt is precisely this suppression that we are proposing to \nuse to quantitatively access saturation effects in the nuclear wavefunction.\nThe purpose of this paper is to provide a realistic estimate of \nthe nuclear suppression in diffractive\ncross sections in a regime that could be measured \nin future nuclear DIS experiments. \n\n\nNuclear DIS data from fixed target experiments, in particular \nE665~\\cite{Adams:1994bw} and NMC~\\cite{Arneodo:1994qb,*Arneodo:1994id} \nhave already been much discussed in the literature as\ndemonstrations of \\emph{color transparency} (see e.g. \nRefs.~\\cite{Frankfurt:1991nx,Frankfurt:1993it,Brodsky:1994kf,Kopeliovich:2001xj,Frankfurt:2005mc}). The form of nuclear modification to the incoherent\ndiffraction in terms of the dipole cross section \nthat we have rederived is not new (see \ne.g.~\\cite{Kopeliovich:1991pu,Kopeliovich:2001xj}).\nSo far, however, less attention\nhas been paid to inelastic diffraction in future DIS experiments.\nThe production \ncross sections have not been calculated using the same\nCGC inspired cross sections that have been used\nsuccessfully to confront HERA data, as we intend to do here.\nIn this work we concentrate on the $J\/\\Psi$ \nbecause its small size means that the interaction\nof the dipole with the target is calculable in weak coupling even at\nsmall $Q^2$.\n\nThe importance of diffraction in understanding gluon saturation has been discussed\nand our basic setup motivated in Ref.~\\cite{Kowalski:2007rw}.\nNuclear modifications \nto the diffractive structure functions, integrated over the momentum transfer\n$t$, were computed in Ref.~\\cite{Kowalski:2008sa}. Vector meson production \nat future DIS experiments was recently \ndiscussed from a more experimental point of view in\nRef.~\\cite{Caldwell:2009ke}, and coherent production cross sections (integrated\nover $t$) calculated in Ref.~\\cite{Goncalves:2009za}.\nAn interesting discussion on coherent and incoherent diffraction\nand gluon saturation in the nucleus can be found in Ref.~\\cite{Tuchin:2008np}.\nIn this study we want to take a step beyond the discussion \nof inclusive diffraction in Refs.~\\cite{Kowalski:2007rw,Kowalski:2008sa}\n to understand the $t$ dependence in more detail.\n\n \n\\section{Dipole cross sections}\n\\label{sec:dipxs}\n\nThere are many dipole cross section parametrizations available \nin the literature, and\nwe have taken for this study two representative samples. One is the \nIIM~\\cite{Iancu:2003ge} dipole cross section, which is a \nparametrization including the most important features \nof BK~\\cite{Balitsky:1995ub,*Kovchegov:1999yj,*Kovchegov:1999ua}\nevolution. The detailed expression for the\ndipole cross section can be found in Ref.~\\cite{Iancu:2003ge};\nwe use here the values of the parameters from the newer\nfit to HERA data including charm~\\cite{Soyez:2007kg}\nthat was also used to compute diffractive structure functions in\nRef.~\\cite{Marquet:2007nf}. We also want to compare\nour results to a parametrization with an eikonalized DGLAP-evolved gluon distribution.\nFor this purpose we will use an approximation of the IPsat dipole \ncross section~\\cite{Kowalski:2003hm,Kowalski:2006hc}. \n\n\nTo extend the dipole cross section from protons to nuclei\n we will take the independent\nscattering approximation that is usually used in Glauber theory \nand write the $S$-matrix as\n\\begin{equation}\\label{eq:sfact}\nS_A({\\mathbf{r}_T},{\\mathbf{b}_T},x) = \\prod_{i=1}^A S_p({\\mathbf{r}_T},{\\mathbf{b}_T}-{\\mathbf{b}_T}_i,x).\n\\end{equation}\nHere we conventionally parametrize the energy dependence of the scattering\namplitude with $x$, the Bjorken variable of the DIS event\\footnote{\nNote that strictly speaking the relation between $x$ and the \nenergy of the dipole-target scattering depends \non $Q^2$, not only $r$. Using $x$ here is justified\nin a high energy approximation\nwhere the energy of the dipole in the target rest frame\nis approximately the same as that of the virtual photon.}.\nThe variables ${\\mathbf{b}_T}_i$ in Eq.~\\nr{eq:sfact} \nare the nucleon coordinates that we will discuss in \nSec.~\\ref{sec:comp}.\nThis independent scattering assumption is\nnatural in IPsat-like parametrizations or the MV~\\cite{McLerran:1994ni} model, \nwhere, denoting $r = |{\\mathbf{r}_T}|,$ $S({\\mathbf{r}_T}) \\sim e^{-r^2 Q_\\mathrm{s}^2\/4}$ with a saturation scale \n$Q_\\mathrm{s}^2$ proportional to the nuclear thickness $T_A(b)$.\nHigh energy evolution, however, introduces an anomalous dimension that leads,\nin the nuclear case, to what could be called leading twist shadowing.\nWith an anomalous dimension\n$S\\sim e^{-(Q_\\mathrm{s} r)^{2\\gamma}}$ with $\\gamma \\neq 1$, a proportionality\n$Q_\\mathrm{s}^2 \\sim T_A(b)$ is not equivalent to Eq.~\\nr{eq:sfact}. A solution to this \nproblem (see also the more detailed discussion in~\\cite{Kowalski:2008sa}) \nwould require a realistic impact parameter dependent solution to the \nBK equation which, we feel fair to say, is not yet available.\nWe point the reader e.g. to Ref.~\\cite{GolecBiernat:2003ym} for a \ndiscussion of the difficulties. These are related to the long distance \nCoulomb tails that, physically, are regulated at the confinement length\nscale that is not enforced in a first principles weak coupling calculation.\nThe effect of BK evolution is important for the CGC description\nof the forward suppression of particle production\nin dAu-collisions at RHIC (for a review see~\\cite{Jalilian-Marian:2005jf}).\nIn our case the difficulty is greater since we are interested\nnot only in the relatively smooth average gluon density, but\nits variations at smaller length scales of the order of the\nproton radius.\nWe thus leave the modifications of Eq.~\\nr{eq:sfact} due to\nthe effects of evolution to a future study.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{amplitude.pdf}\n\\caption{\nThe $r$-dependence of the different proton dipole cross sections used,\nat $x=0.0001$ and $b=0$.\nAs discussed in Sec.~\\ref{sec:res}, the ``IPnonsat''-curve is\nEq.~\\nr{eq:BEKWfact} linearized in $r^2F(x,r)$.\n} \\label{fig:sigmap}\n\\end{figure}\n\nThe IIM parametrization assumes, either explicitly or implicitly,\na factorizable ${\\mathbf{b}_T}$ dependence \n\\begin{eqnarray}\\label{eq:factbt}\n{\\frac{\\ud \\sigma^\\textrm{p}_\\textrm{dip}}{\\ud^2 \\bt}}({\\mathbf{b}_T},{\\mathbf{r}_T},x) &=& 2 \\left( 1 - S_p({\\mathbf{r}_T},{\\mathbf{b}_T},x)\\right)\n\\\\ \\nonumber\n&=& 2 \\,T_p({\\mathbf{b}_T}) {\\mathcal{N}}(r,x),\n\\end{eqnarray}\nWe take, following Ref.~\\cite{Marquet:2007nf}, a Gaussian profile\n$T_p({\\mathbf{b}_T}) = \\exp\\left(-b^2\/2 B_p\\right)$ with \n$B_p=5.59\\ \\textrm{GeV}^{-2}$ (see Sec.~\\ref{sec:res} for a discussion of \nthis largish numerical value). \n\nIn the IPsat model the impact parameter dependence is\nincluded in the saturation scale as\n\\begin{equation}\\label{eq:unfactbt}\n{\\frac{\\ud \\sigma^\\textrm{p}_\\textrm{dip}}{\\ud^2 \\bt}}({\\mathbf{b}_T},{\\mathbf{r}_T},x)\n = 2\\,\\left[ 1 - \\exp\\left(- r^2 F(x,r) T_p({\\mathbf{b}_T})\\right) \n\\right].\n\\end{equation}\nHere $T_p({\\mathbf{b}_T}) = \\exp\\left(-b^2\/2 B_p\\right)$ \nis the impact parameter profile function in the proton \nwith $B_p=4.0\\ \\textrm{GeV}^2$ and $F$ is proportional to the \nDGLAP evolved gluon distribution~\\cite{Bartels:2002cj}\n\\begin{equation}\nF(x,r^2) = \n\\frac{1}{2 \\pi B_p}\n\\frac{ \\pi^2 }{2 {N_\\mathrm{c}}} \\alpha_{\\mathrm{s}} \\left(\\mu_0^2 + \\frac{C}{r^2} \\right) \nx g\\left(x,\\mu_0^2 + \\frac{C}{r^2} \\right), \n\\label{eq:BEKW_F}\n\\end{equation}\nwith $C$ chosen as 4 and $\\mu_0^2=1.17\\ \\textrm{GeV}^2$ resulting from the \nfit~\\cite{Kowalski:2006hc}. The proton dipole cross sections used are\nplotted in Fig.~\\ref{fig:sigmap} for $x=0.0001$.\n\nWe would generally prefer the unfactorized $b$-dependence\nof Eq.~\\nr{eq:unfactbt} to the factorized one in Eq.~\\nr{eq:factbt}\nbecause it allows for the correct unitarity\nlimit of the scattering amplitude at all impact parameters \n(see the discussion in Ref.~\\cite{Kowalski:2008sa}). \nHowever, there seems to be no clear difference between the two in\nterms of the quality of the description of HERA data, and for the sake \nof computational simplicity we will in this work limit ourselves to\nthe factorized dependence and approximate the IPsat dipole cross section\nby\n\\begin{equation}\\label{eq:BEKWfact}\n{\\frac{\\ud \\sigma^\\textrm{p}_\\textrm{dip}}{\\ud^2 \\bt}}({\\mathbf{b}_T},{\\mathbf{r}_T},x)\n \\approx 2 T_p({\\mathbf{b}_T}) \\,\\left[ 1 - \\exp\\left(- r^2 F(x,r)\\right)\n\\right]\n\\end{equation}\nusing the same $F(x,r)$ defined in Eq.~\\nr{eq:BEKW_F}. This approximation\nbrings the IPsat parametrization to the form Eq.~\\nr{eq:factbt}\nwith ${\\mathcal{N}}(r,x)=\\left[ 1 - \\exp\\left(- r^2 F(x,r)\\right)\\right]$;\nin fact this is the form used already in Ref.~\\cite{Bartels:2002cj}; we however\nuse the gluon distribution from the IPsat fit~\\cite{Kowalski:2006hc} \nfor convenience. Improving this\ndescription goes hand in hand with giving up the approximation\nof independent scatterings off the nucleons, Eq.~\\nr{eq:sfact},\nand is left for future work. As we shall see in the following, these \napproximations enable us to write the cross section for incoherent\ndiffraction in a form which is much simpler to evaluate numerically \nthan one with a general $b$-dependence.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{totxs-boosted.pdf}\n\\caption{Comparison of the used dipole cross sections to HERA \ndata~\\cite{Chekanov:2004mw,*Aktas:2005xu} on diffractive vector meson production.\n} \\label{fig:hera}\n\\end{figure}\n\n\n\n\\section{Computing diffractive cross sections}\n\\label{sec:comp}\n\n\n\nThe cross section for quasielastic vector meson \nproduction in nuclear DIS is\n\\begin{equation} \\label{eq:xsec}\n\\frac{\\, \\mathrm{d} \\sigma^{\\gamma^* A \\to V A }}{\\, \\mathrm{d} t} \n= \\frac{R_g^2(1+\\beta^2)}{16\\pi} \\Aavg{|{\\mathcal{A}}({x_\\mathbb{P}},Q^2,{\\boldsymbol{\\Delta}_T})|^2}.\n\\end{equation}\nwith $t=-{\\boldsymbol{\\Delta}_T}^2$. \nThe dipole cross section is evaluated at the \nenergy scale corresponding to the rapidity gap between the \nvector meson and the target ${x_\\mathbb{P}}$.\nTo translate this into the photon-target center of mass energy $W$ \nthat is often used to present experimental results note that \n${x_\\mathbb{P}} = (M_{J\/\\Psi}^2 + Q^2)\/(W^2 + Q^2)$.\nThe factor $1+\\beta^2$ accounts for the\nreal part of the scattering amplitude and the factor $R_g^2$ corrects\nfor the skewedness effect, i.e. that the gluons in the target are probed at \nslightly different $x$~\\cite{Shuvaev:1999ce,*Martin:1999wb}. \nFor these corrections\nwe follow the prescription of Ref.~\\cite{Watt:2007nr}, taking them \nas\n\\begin{eqnarray}\n\\beta &=& \\tan \\frac{\\pi \\lambda}{2}\n\\\\\nR_g &=& \\frac{2^{2 \\lambda+3}}{\\sqrt{\\pi}}\\frac{\\Gamma(\\lambda + 5\/2)}{\\lambda+4} \n\\quad \\textrm{ with}\n\\\\\n\\lambda &=& \\frac{\\partial \\ln {\\mathcal{A}}_{T,L}^{\\gamma^*p\\to J\/\\Psi p}}{\\partial \\ln 1\/{x_\\mathbb{P}}}.\n\\end{eqnarray}\nThese corrections depend, in general, on $t$, which we take into account in our calculation. \nFor the full IPsat model $\\lambda$ changes by about 5\\% between $t=0$ and \n$-t=0.5\\ \\textrm{GeV}^2$. For the factorized impact parameter dependence in \nEqs.~\\nr{eq:factbt} and~\\nr{eq:BEKWfact} $\\lambda$ is \nindependent of $t$.\nWe calculate the correction terms from the energy dependence of the nucleon\nscattering amplitudes and use the same values for the nucleus at the \nsame $Q^2,{x_\\mathbb{P}}$. Since the difference in $\\lambda$ extracted from the nucleus \nand the nucleon cross sections is small (compared to the value of $\\lambda$) and \n$R_g$ and $\\beta$ are in themselves corrections to the cross section, this \napproximation is justified. In addition this approximation has the advantage that \nthese corrections cancel on the nucleus\/nucleon cross section ratio.\nThe real part and skewedness corrections, especially $R_g$ are, however, a significant \nfactor in the absolute normalization of the cross section and are \nnecessary for the agreement with HERA data.\n\nThe imaginary part of the scattering amplitude is the Fourier-transform of the\ndipole cross section from ${\\mathbf{b}_T}$ to ${\\boldsymbol{\\Delta}_T}$ contracted with the \noverlap between the vector meson and virtual photon wave functions:\n\\begin{multline}\\label{eq:ampli}\n{\\mathcal{A}}({x_\\mathbb{P}},Q^2,{\\boldsymbol{\\Delta}_T}) \n= \\int \\, \\mathrm{d}^2 {\\mathbf{r}_T} \\int \\frac{\\, \\mathrm{d} z}{4\\pi} \\int \\, \\mathrm{d}^2 {\\mathbf{b}_T} \n\\\\\n\\times [\\Psi_V^* \\Psi](r,Q^2,z)\ne^{-i {\\mathbf{b}_T} \\cdot {\\boldsymbol{\\Delta}_T}} \n{\\frac{\\ud \\sigma_\\textrm{dip}}{\\ud^2 \\bt}}({\\mathbf{b}_T},{\\mathbf{r}_T},{x_\\mathbb{P}}),\n\\end{multline}\nwhere we have followed the normalization convention of~\\cite{Kowalski:2006hc}.\nFor the virtual photon--vector meson wavefunction overlap we\nuse the ``boosted Gaussian'' parametrization from Ref.~\\cite{Kowalski:2006hc}.\nWe have also tested the ``gaus-LC'' wavefunction also used in \nRef.~\\cite{Kowalski:2006hc}. Although the ``boosted Gaussian'' seems preferred\nby HERA data, also the ``gaus-LC'' parametrization is compatible with \nthe data within the experimental errors. The cross sections for the\nproton differ by factors of the order of 10\\%.\nThe interaction of the gluon target with the dipole can in general depend also on \n${\\boldsymbol{\\Delta}_T}$, which introduces terms that couple ${\\mathbf{r}_T},$ ${\\boldsymbol{\\Delta}_T}$ and $z$ in \nEq.~\\nr{eq:ampli}. For the $J\/\\Psi$ and the range in $t$ considered in this paper\n${\\boldsymbol{\\Delta}_T}$ is sufficiently small compared to the relevant values of $1\/r$ that \nwe can neglect this coupling, which simplifies the structure considerably.\nLighter vector mesons would require a more general treatment.\n\nThe average over the positions of the nucleon in the nucleus is denoted here by\n\\begin{equation} \\label{eq:aavg}\n\\Aavg{\\mathcal{O}(\\{ {\\mathbf{b}_T}_i \\})} \n\\equiv \\int \\prod_{i=1}^{A}\\left[ \\, \\mathrm{d}^2 {\\mathbf{b}_T}_i T_A({\\mathbf{b}_T}_i) \\right] \n\\mathcal{O}(\\{ {\\mathbf{b}_T}_i \\}).\n\\end{equation}\nHere $T_A$ is the Woods-Saxon distribution with nuclear radius \n$R_A = (1.12 A^{1\/3}-0.86 A^{-1\/3})\\ \\textrm{fm} $ and surface thickness \n$d=0.54\\ \\textrm{fm}$.\nThis expectation value is equivalent to the average\nover nucleon configurations in a Monte Carlo Glauber \ncalculation.\nWe are assuming that the positions ${\\mathbf{b}_T}_i$ are independent, i.e. \nneglecting nuclear correlations that would be a subject of\ninterest in their own right (see e.g.~\\cite{Alvioli:2009ab}).\nThe coherent cross section is obtained by averaging the amplitude\nbefore squaring it, $|\\Aavg{{\\mathcal{A}}}|^2$, and the incoherent one is\nthe variance $\\Aavg{|{\\mathcal{A}}|^2} - |\\Aavg{{\\mathcal{A}}}|^2$ that measures the fluctuations\nof the gluon density inside the nucleus. Because\n$\\Aavg{{\\mathcal{A}}}$ is a very smooth function of ${\\mathbf{b}_T}$, its Fourier transform \nvanishes rapidly for $\\Delta \\gtrsim 1\/R_A$. Therefore at large $\\Delta$\nthe quasielastic cross section \\nr{eq:xsec} is almost purely incoherent.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{coherent_q0.pdf}\n\\caption{The quasielastic and coherent diffractive $J\/\\Psi$ cross sections in gold\nnuclei at $Q^2= 0$ and ${x_\\mathbb{P}} = 0.001$.\nShown are \nthe IPsat and IIM parametrizations. We also show the \nresult for the linearized ``IPnonsat'' version \n(used e.g. in Ref.~\\cite{Caldwell:2009ke}) where the incoherent\ncross section is explicitly $A$ times that of the proton. \nOur approximation \\nr{eq:amplisq} \nis not valid for small $|t|$; the corresponding part of the distribution\nhas been left out.\n} \\label{fig:dsigmavst}\n\\end{figure}\n\n\nThe cross section for quasielastic vector meson production is now expressed\nin terms of the dipole scattering amplitude as\n\\begin{multline} \\label{eq:incxsec}\n\\frac{\\, \\mathrm{d} \\sigma^{\\gamma^* A \\to V A^* }}{\\, \\mathrm{d} t} \n= \n\\frac{R_g^2(1+\\beta^2)}{16\\pi} \n\\int \n\\frac{\\, \\mathrm{d} z}{4 \\pi} \\frac{\\, \\mathrm{d} z'}{4 \\pi}\n\\, \\mathrm{d}^2 {\\mathbf{r}_T} \\, \\mathrm{d}^2 {\\mathbf{r}_T}'\n\\\\ \\times\n\\left[ \\Psi^*_V \\Psi \\right] (r,z,Q)\n\\, \\left[ \\Psi^*_V \\Psi \\right](r',z',Q)\n\\\\ \\times\n\\Aavg{ \\left| \\mathcal{A}_{q\\bar{q}}\\right|^2({x_\\mathbb{P}},r,r',{\\boldsymbol{\\Delta}_T}) } \\, .\n\\end{multline}\nWe now average the square of the dipole scattering amplitude over the \nnucleon coordinates, using the assumptions of\nEqs.~\\nr{eq:sfact} and~\\nr{eq:factbt} and taking the large $A$ limit.\nWe are additionally assuming that $T_A$ is a smooth function on the \ndiscance scale defined by $B_p$.\nAveraging the square of the amplitude gives the total quasielastic \ncontribution, but we only keep the terms \nthat contribute at large $|t| \\gg 1\/R_A^2$, which leaves us \nwith the expression \n\\begin{multline}\\label{eq:amplisq}\n\\left| \\mathcal{A}_{q\\bar{q}}\\right|^2({x_\\mathbb{P}},r,r',{\\boldsymbol{\\Delta}_T}) \n=\n16 \\pi B_p \\int \\, \\mathrm{d}^2 {\\mathbf{b}_T} \\sum_{n=1}^A\n\\frac{1}{n} \\binom{A}{n} \n\\\\ \\times \ne^{-B_p {\\boldsymbol{\\Delta}_T}^2\/n}\ne^{-2 \\pi B_p A T_A(b)\n\\left[ {\\mathcal{N}}(r) + {\\mathcal{N}}(r') \\right] } \n\\\\ \\times\n\\left( \\frac{\\pi B_p {\\mathcal{N}}(r){\\mathcal{N}}(r') T_A(b) }\n {1 - 2 \\pi B_p T_A(b)\\left[ {\\mathcal{N}}(r) + {\\mathcal{N}}(r') \\right] } \\right)^n\n.\n\\end{multline}\nNote that Eqs.~\\nr{eq:sfact} and~\\nr{eq:factbt} have enabled us to\nwrite the leading contributions as proportional to the\n(Gaussian) proton impact parameter profile, which can then be \nFourier-transformed analytically. Giving up either of these approximations\nwould force us to numerically Fourier-transform the ``lumpy'' \n$b$-dependence corresponding to a fixed configuration\nof the nucleon positions. This would make the numerical calculation \nmuch more demanding and is left for future work. \n\nThe terms with $n \\geq 2$ correspond to scattering off a system of several \noverlapping nucleons simultaneously, leading to slower suppresion with $|t|$.\nIn practice we have verified numerically that they do not\ncontribute to our results at the values of $t$ we are interested in \n(the $n=2$ contribution is typically $\\lesssim 2\\%$ of the $n=1$-one,\nonly reaching $5\\%$ at $-t\\gtrsim 0.5 \\ \\textrm{GeV}^2$ ) and \nwill neglect them in the following. This leaves us with the expression\n\\begin{multline}\\label{eq:amplisqn1}\n\\left| \\mathcal{A}_{q\\bar{q}}\\right|^2({x_\\mathbb{P}},r,r',{\\boldsymbol{\\Delta}_T}) \n=\n16 \\pi B_p A \\int \\, \\mathrm{d}^2 {\\mathbf{b}_T} \n\\\\ \\times \ne^{-B_p {\\boldsymbol{\\Delta}_T}^2}\ne^{-2 \\pi B_p A T_A(b)\n\\left[ {\\mathcal{N}}(r) + {\\mathcal{N}}(r') \\right] } \n\\\\ \\times\n\\left( \\frac{\\pi B_p {\\mathcal{N}}(r){\\mathcal{N}}(r') T_A(b) }\n {1 - 2 \\pi B_p T_A(b)\\left[ {\\mathcal{N}}(r) + {\\mathcal{N}}(r') \\right] } \\right).\n\\end{multline}\nEquation \\nr{eq:amplisqn1} has a very clear interpretation. The \nsquared amplitude is proportional to $A$ times the squared amplitude\nfor scattering off a proton, corresponding to the dipole scattering \nindependently off the nucleons in a nucleus. This sum of independent\nscatterings is then multiplied by a nuclear attenuation factor\n\\begin{multline}\n\\frac{ e^{-2 \\pi B_p A T_A(b) \\left[ {\\mathcal{N}}(r) + {\\mathcal{N}}(r') \\right] } } \n {1 - 2 \\pi B_p T_A(b)\\left[ {\\mathcal{N}}(r) + {\\mathcal{N}}(r') \\right] } \n\\approx\n\\\\\n e^{-2 \\pi (A-1) B_p T_A(b) \\left[ {\\mathcal{N}}(r) + {\\mathcal{N}}(r') \\right] } ,\n\\end{multline}\nwhich accounts for the requirement that the dipole must \\emph{not}\nscatter inelastically off the other $A-1$ nucleons in the target (otherwise the\ninteraction would not be diffractive). \nNote that the factor \n$4 \\pi B_p {\\mathcal{N}}(r,{x_\\mathbb{P}})={ \\sigma^\\textrm{p}_\\textrm{dip} }(r,{x_\\mathbb{P}})$ is the proton-dipole cross section for a\ndipole of size $r$. Thus this attenuation corresponds to the\nprobability of a dipole with a cross section which is the average \nof dipoles with $r$ and $r'$ to pass though the nucleus.\nA similar expression \ncan be found e.g. in Ref.~\\cite{Kopeliovich:2001xj}.\n\n\n\nFor comparison, the coherent cross section in our approximation is given by\n\\begin{equation} \\label{eq:coh}\n\\frac{\\, \\mathrm{d} \\sigma^{\\gamma^* A \\to V A }}{\\, \\mathrm{d} t} \n=\\frac{R_g^2(1+\\beta^2)}{16\\pi} \\left| \\Aavg{{\\mathcal{A}}({x_\\mathbb{P}},Q^2,{\\boldsymbol{\\Delta}_T})} \\right|^2,\n\\end{equation}\nwhere in the large $A$ and smooth nucleus limit the amplitude is\n\\begin{multline}\\label{eq:cohampli}\n\\Aavg{{\\mathcal{A}}({x_\\mathbb{P}},Q^2,{\\boldsymbol{\\Delta}_T}) }\n= \\int \\frac{\\, \\mathrm{d} z}{4\\pi} \\, \\mathrm{d}^2 {\\mathbf{r}_T} \\, \\mathrm{d}^2 {\\mathbf{b}_T} e^{-i {\\mathbf{b}_T} \\cdot {\\boldsymbol{\\Delta}_T}} \n\\\\\n \\times [\\Psi_V^*\\Psi](r,Q^2,z)\n\\, 2 \\left[ 1-\\exp\\left\\{ - 2 \\pi B_p A T_A(b) {\\mathcal{N}}(r,{x_\\mathbb{P}}) \\right\\} \\right].\n\\end{multline}\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{plot_Q.pdf}\n\\caption{The ``nuclear transparency'' \nratio of cross sections vs. $Q^2$ for IPsat, IIM \nparametrizations at ${x_\\mathbb{P}}= 10^{-2}$ (the upper three curves, blue)\nand $10^{-4}$ (the lower 3 curves, black).\nFor comparison we also include\nwe also include the result if unitarization effects are included \nat the nucleus but not at the nucleon level in the IPsat-parametrization. \n(See text for discussion).\n}\\label{fig:ratiovsq}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{plot_Q_gaus-lc.pdf}\n\\caption{The ``nuclear transparency'' \nratio of cross sections vs. $Q^2$ using the ``Gaus-LC'' vector meson \nwavefunctions. The labeling is the same as in Fig.~\\ref{fig:ratiovsq}.\n}\\label{fig:ratiovsq_gauslc}\n\\end{figure}\n\n\n\n\n\\section{Results and discussion}\n\\label{sec:res}\n\nWe first test our dipole cross section parametrizations and vector meson wave\nfunctions by comparing them to HERA results~\\cite{Chekanov:2004mw,*Aktas:2005xu} \non diffractive $J\/\\Psi$ production\nthat is known to be well described by dipole model \nfits~\\cite{Kowalski:2006hc,Marquet:2007qa}. The comparison is quite satisfactory, as\ncan be seen from Fig.~\\ref{fig:hera}. In addition to the factorized \napproximation (Eq.~\\nr{eq:BEKWfact}, ``factorized IPsat'' in the figure) that \nwe are using in the rest of this paper, also shown is the result with the original\nIPsat parametrization (Eq.~\\nr{eq:unfactbt}, denoted ``IPsat'' in the figure). The \nfactorized approximation differs from the original one slightly at small \n$Q^2$, but the difference is not significant for our purpose of establishing a \nreasonable baseline for computing nuclear effects.\n\nWe note here that the diffractive slope parameters\nin the parametrizations are different, $B_p=4.0\\ \\textrm{GeV}^{-2}$ for IPsat and \n$B_p= 5.59\\ \\textrm{GeV}^{-2}$\nfor IIM; since these are correlated with the other parameters in the fits leading to\nthe parameter values used we do not wish to alter them here.\nOur approximation of a factorized $b$-dependence with a constant $B$ \ndoes not allow us to describe the observed weak energy and $Q^2$ dependence of the\ndiffractive slope. \nThe larger $B$ that we use for IIM comes from the $\\sigma_0$ normalization\nin a fit to inclusive $F_2$ data, and also agrees with the observed slopes in \ninclusive diffraction at large $\\beta$ and small \n${x_\\mathbb{P}}$~\\cite{Chekanov:2004hy,*Aktas:2006hx} and exclusive $\\rho$ and $\\phi$\ndata~\\cite{Adloff:1999kg,*Chekanov:2005cqa}. The HERA $J\/\\Psi$-data,\non the other hand, has a smaller slope \n$\\sim 4\\ \\textrm{GeV}^{-2}$~\\cite{Chekanov:2004mw,*Aktas:2005xu}. \nThe $t$-slope in\nthe IPsat parametrization is mostly determined by this $J\/\\Psi$-measurement,\nand an agreement with the larger measured slopes for $\\rho$ and $\\phi$\nis obtained by taking into account the larger size of the wavefunctions\nof these lighter mesons.\n\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{plot_x.pdf}\n\\caption{The ``nuclear transparency'' ratio of cross sections vs. ${x_\\mathbb{P}}$ \nusing the IPsat and IIM parametrizations for $Q^2=0$ and $Q^2=10\\ \\textrm{GeV}^2$.\n}\\label{fig:ratiovsx}\n\\end{figure}\n\n\n\nThe differential cross section $\\, \\mathrm{d} \\sigma^{\\gamma^* A \\to J\/\\Psi A}\/\\, \\mathrm{d} t$\n for $A=197$ (gold) as a function of $t$ is presented in Fig.~\\ref{fig:dsigmavst}. \nWe show the cross \nsection at ${x_\\mathbb{P}} = 0.001$ \nfor photoproduction. As we performed\nthe nuclear wavefunction average leading to Eq.~\\nr{eq:amplisq} in the \napproximation where $|t|$ is large, neglecting the coherent contribution,\nwe cannot extend our incoherent curves to small $|t|$. \nFor comparison we show the corresponding\n``IPnonsat'' result where the IPsat model is linearized in $r^2 F(x,r)$. \nThis curve corresponds to the calculation done in Ref.~\\cite{Caldwell:2009ke},\nincluding both the coherent and incoherent contributions, but without the effect\nof multiple scattering off different nucleons (i.e. the incoherent cross section is\nexplicitly $A$ times the one for a proton). \nAs one can see, the nuclear modification\ndue to multiple scattering (resulting mostly from the\nfactor $e^{-2 \\pi B_p A T_A(b) \\left[ {\\mathcal{N}}(r) + {\\mathcal{N}}(r') \\right] } $ in \nEq.~\\nr{eq:amplisq}) is very large. \n In the full black disk limit \nof ${\\mathcal{N}}(r)=1$ this factor becomes $\\approx e^{-0.5 A^{1\/3}}$ and completely\nsupresses the contribution from the center of a large nucleus, leaving only an area\nof $\\approx 2 \\pi d R_A \\sim A^{1\/3}$ contributing to the integral over ${\\mathbf{b}_T}$.\nThus the cross section in \nthe black disc limit behaves as $\\sim A^{1\/3}$ compared to $\\sim A$ in the dilute\nlimit, so a large suppression is to be expected.\n\n\nWe also show in Fig.~\\ref{fig:dsigmavst} the coherent cross sections \n(using Eq.~\\nr{eq:cohampli}). They are also suppressed compared to the linearized \nversion (IPnonsat), but not by as much as the incoherent one. In the linearized\nversion (as can be seen explicitly in Ref.~\\cite{Caldwell:2009ke} where this case\nwas considered) the ratio between the coherent cross section at $t=0$ and the incoherent\none extrapolated to $t=0$ is $A$. In the IPsat model we get\n$270$ ($250$) and in the IIM model $300$ ($270$) at $Q^2=0$ ($Q^2=10\\ \\textrm{GeV}^2$).\nThis would make it slightly easier to measure the first \ndiffractive dip in the coherent cross section, since the background from the incoherent\nprocess is smaller by a factor of 2 than the linearized \nestimate~\\cite{Caldwell:2009ke}.\n\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{ratio_Q.pdf}\n\\caption{\nThe incoherent cross section integrated over the interval $0.1\\ \\textrm{GeV}^2< -t < 0.3 \\ \\textrm{GeV}^2$\ndivided by the coherent cross section integrated over $0 < -t < 0.1 \\ \\textrm{GeV}^2$\nas a function of $Q^2 + M_{J\/\\Psi}^2$.\n} \\label{fig:incohovercohQ}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{ratio_x.pdf}\n\\caption{\nThe incoherent cross section integrated over the interval $0.1\\ \\textrm{GeV}^2< -t < 0.3 \\ \\textrm{GeV}^2$\ndivided by the coherent cross section integrated over $0 < -t < 0.1 \\ \\textrm{GeV}^2$\nas a function of ${x_\\mathbb{P}}$.\n} \\label{fig:incohovercohx}\n\\end{figure}\n\n\nTo demonstrate the nuclear dependence further we show in Fig.~\\ref{fig:ratiovsq}\nthe ratio of the cross section in a gold nucleus to that in a nucleon as a \nfunction of $Q^2$. Historically this ratio is known as the ``nuclear transparency''.\nIts smallness at low energy, similarly to coresponding quantities in \nhadron-nucleus scattering, is due to the interactions of the $J\/\\Psi$ as it propagates\nthrough the nucleus. The growth of the transparency towards $1$ for increasing\n$Q^2$~\\cite{Adams:1994bw,Arneodo:1994qb,*Arneodo:1994id} is \na demonstration of \\emph{color transparency} (see e.g. \nRef.~\\cite{Frankfurt:1991nx,Frankfurt:1993it,Brodsky:1994kf,Kopeliovich:2001xj,Frankfurt:2005mc,Miller:2010eh}),\nnamely that at large\n$Q^2$ the interacting components of the photon wavefunction are of smaller\nsize $r$ and interact weakly. In our framework color transparency is\nautomatically present in the fact that the dipole cross section approaches zero\nfor $r\\to0$. In Fig.~\\ref{fig:ratiovsq} we also show the result (labeled\n``IPsat, nonsatp'') of using a nonsaturated dipole-nucleon cross section\nin Eq.~\\nr{eq:amplisq}. This corresponds to including \nunitarity effects at the nucleus level but not \nfor a single nucleon. The observed nuclear suppression in this unphysical \nscenario is significantly larger than for the saturated full IPsat\nparametrization, showing the sensitivity of the nuclear transparency\nto saturation effects already at the proton level.\n\nThe IIM parametrization has a much larger nuclear \nsuppression in incoherent diffraction, with the nuclear transparency\nratio close that of an unsaturated dipole-proton cross section.\nTo put this in perspective recall that both parametrizations\ngave an equally good description of the elastic cross section\nmeasured at HERA (Fig.~\\ref{fig:hera}). Since IIM does this with a\nlarger $B_p$ than IPsat, we can infer that the typical ${\\mathcal{N}}$ is smaller, \nso that the elastic cross section $\\sigma^\\textrm{el}\n\\sim B_p {\\mathcal{N}}^2$ is of the same order. \nThe nuclear transparency ratio, on the other hand, depends on the\ntotal dipole-nucleon cross section \n$\\sim B_p {\\mathcal{N}} \\sim \\sigma^\\textrm{el}\/{\\mathcal{N}}$ which is thus \nlarger for IIM. Thus we have a situation where both parametrizations\nhave been fitted to inclusive $F_2$ data\\footnote{Although we have here \napproximated the original IPsat parametrization by factorizing the $b$-dependence.},\nreproduce well the HERA $J\/\\Psi$ cross section, but differ in their\nresult for incoherent diffraction in nuclei. This stresses the importance of\nperforming a global analysis of both inclusive and diffractive data\nto constrain the dipole cross sections, and demonstrates\nthe utility of eventual incoherent diffractive measurements in such \nan analysis.\n\nFigure \\ref{fig:ratiovsq_gauslc} shows the same $Q^2$-dependence\nusing the ``gaus-LC'' wavefunction. It puts more\nweight on large dipole sizes, leading to a stronger nuclear suppression. \nThe cross section ratio typically decreases by $\\sim 0.04$ from \nthe ``boosted Gaussian'' wavefunction, but the\nrelative structure between the different dipole cross sections stays the same.\nThe difference between the cross sections themselves is larger,\nbut much of the it cancels in the ratio. The existing HERA data \nis not precise enough to fully discriminate between different models for\nthe vector meson wavefunction, a situation which should also improve\nwith planned new DIS experiments.\n\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{cohincoh.pdf}\n\\caption{\nThe ratio of the coherent (at $t=0$) and incoherent (at $t=-0.5\\ \\textrm{GeV}^2$,\nbut in our approximation this does not depend on $t$) \ncross sections to the corresponding ones for a proton; normalized with \n$A^2$ and $A$ respectively. Plotted as a function of $A$,\nfor ${x_\\mathbb{P}} = 0.001$ and $Q^2=10\\ \\textrm{GeV}^2$.\n} \\label{fig:cohincoh}\n\\end{figure}\n\n\n\nThe energy dependence of the nuclear suppression (again for $A=197$)\nis shown in Fig.~\\ref{fig:ratiovsx} for both IPsat and IIM parametrizations \nat $Q^2=0$ and $Q^2=10\\ \\textrm{GeV}^2$. Again \nwe see the larger nuclear suppression in the IIM model than in IPsat.\nThe differences in the energy (i.e. ${x_\\mathbb{P}}$) dependence of\nthe two dipole cross sections are more clearly visible in the \nphotoproduction result. This is natural, since in the IPsat model \nthe energy dependence at the initial scale of the DGLAP evolution\n(probed at smaller $Q^2$) is almost flat, in stark contrast to the\ntypical behavior resulting from BK evolution. At higher $Q^2$\nthe difference in the $x$-dependence is smaller, although \nthere the IPsat-model, driven by the DGLAP evolution, \nturns over to a \\emph{faster}\nenergy dependence. We have not extrapolated our curves to higher \nenergies, since there is no prospect of experimental measurements. One does \nhowever see from Fig.~\\ref{fig:ratiovsx} that the curves\ncontinue to go down when extrapolated to smaller ${x_\\mathbb{P}}$. This is to be \nexpected since, as discussed previously, one has not yet reached the \nblack disk limit. \n\nIn a realistic experimental setup it might be possible to detect \nor veto the nuclear breakup even when the momentum transfer\n$t$ is not measured very accurately. In this case it will be interesting \nto understand how the relative magnitudes of the incoherent and coherent\ncross sections behave as a function of $Q^2$ and ${x_\\mathbb{P}}$. Generally when approaching \nthe black disk limit the coherent cross section increases and the incoherent one\ndecreases. The relative change shows, however, a smaller dependence on $Q^2$ and \n${x_\\mathbb{P}}$ than the nucleus\/nucleon cross section ratio. This is shown in our \nparametrization in Figs.~\\ref{fig:incohovercohQ} and~\\ref{fig:incohovercohx}, where\nwe plot the \nthe incoherent cross section integrated over the interval $0.1\\ \\textrm{GeV}^2< -t < 0.3 \\ \\textrm{GeV}^2$\ndivided by the coherent cross section integrated over $0 < -t < 0.1 \\ \\textrm{GeV}^2$\nas a function of $Q^2 + M_{J\/\\Psi}^2$ and ${x_\\mathbb{P}}$.\nFigure \\ref{fig:cohincoh} further demonstrates the relative similarity of the \nnuclear suppression in the coherent and incoherent cross sections. \nShown is the $A$ dependence of the ratios \n$ (\\, \\mathrm{d}\\sigma^A_\\mathrm{incoh}\/\\, \\mathrm{d} t)\/(A \\, \\mathrm{d} \\sigma^p\/\\, \\mathrm{d} t)$\n(which, in our approximation, is independent of $t$)\nand \n$\\left. (\\, \\mathrm{d}\\sigma^A_\\mathrm{coh}\/\\, \\mathrm{d} t)\/(A^2 \\, \\mathrm{d} \\sigma^p\/\\, \\mathrm{d} t)\\right|_{t=0}$\nfor $Q^2=10\\ \\textrm{GeV}^2$ and ${x_\\mathbb{P}}=0.001$. Note that the coherent and the incoherent \ncross sections are normalized by different powers of $A$ and that\nwidth of the coherent peak at small $t$ also depends on $A$.\n\nFigures \\ref{fig:dsigmavst} and \\ref{fig:ratiovsq} are our main result.\nOur calculation uses as input only well tested parametrizations that have been fit\nto existing HERA data and nuclear geometry. We work strictly in the\nsmall $x$-limit which makes our formalism simple and transparent.\nThis paper provides realistic estimates for the absolute cross sections \nthat could be measured in future nuclear DIS experiments. \nWe have, however, made several simplifying assumptions in our calculation, the most \nimportant being a) the factorized impact parameter dependence Eq.~\\nr{eq:factbt},\nb) the assumption of independent scattering off different nucleons\nEq.~\\nr{eq:sfact} and c) neglecting nucleon-nucleon correlations. Including\nthese effects in a physically correct manner and discussing how they could be \nstudied experimentally is left for future work. As can be seen from \nthe values of the nuclear suppression in Figs.~\\ref{fig:ratiovsq} and~\\ref{fig:ratiovsx},\nthe effects of high densities, gluon saturation and unitarity on the\nincoherent cross section are large.\nThus incoherent diffraction in future nuclear DIS experiments will\nbe a sensitive probe of small-$x$ physics.\n\n\n\n\\section*{Acknowledgements}\nWe thank E. Aschenauer, H. Kowalski and W. Horowitz for discussions and K.~J.~Eskola\nfor a careful reading of the manuscript.\nThe work of T.L. has been supported by the Academy of Finland, project \n126604. T.L. wishes to thank the INT at the \nUniversity of Washington for \nits hospitality during the completion of this work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}