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+{"text":"\\section{Introduction}\n\nThe paper is written for the special issue dedicated to the outstanding physicist Mark Azbel. It addresses the problems of quantum physics, to which Prof. Azbel made a number of seminal \ntheoretical contributions \\cite{AzbelP,AzbelN,Azbel}. \n\nThe angle variable $\\varphi$ is ubiquitous in classical  and  quantum physics. Among  examples of this variable are the rotation angle of the plane rotator  (mechanical pendulum  as a particular case) and the phase difference across  the single Josephson junction (JJ).  The  canonically  conjugate variable to the phase is the angular momentum $M$ (further called moment) in the first example and  the charge  in the second one. \n\nThe history of using the canonically conjugate  pair  ``angle  (phase)--moment'' in quantum mechanics is full of controversies and disputes. In particular, the commutation  relation \n\\begin{equation}\n[\\hat \\varphi,\\hat M]=i\\hbar\n    \\ee{ComS}\nintroduced by \\citet{dirac} was challenged \\cite{judge,PhasRev,Pegg}. Here  $\\hat \\varphi$ and $\\hat  M$ are operators of the angle  (phase) and  the moment, respectively. The problem with the commutation relation was connected with  the non-Hermitian character  of  the phase operator. It was suggested to repair this mostly mathematical flaw  by rewriting the commutation relation in the terms of Hermitian operators  $\\sin \\hat \\varphi$ and $\\cos \\hat \\varphi$ \\cite{suss}. The uncertainty relation \n\\begin{equation}\n\\Delta M \\Delta \\varphi > {\\hbar\\over 2} \n   \\ee{unc}\nwas also under scrutiny \\cite{judge,PhasRev,Pegg}.  Here  $\\Delta M$ and $ \\Delta \\varphi$ are uncertainties of the moment and the phase, respectively.\n \nAnother problem  (but connected with the first one) is that the phase  $\\varphi$ is defined modulo $2\\pi$. One can choose the phase defined in  an   interval from an arbitrarily chosen phase  $\\varphi_0$  to $\\varphi_0  +2\\pi$  (compact phase),  or  the phase ranging from $-\\infty$ to $\\infty$ (extended phase). If  the  phases differing by the integer number $s$ of  $2\\pi$ describe the same state, it does not matter  at all which phase, compact or extended,  one uses in  the theoretical analysis.  The analysis (if correct) must lead to the same results. However, in quantum mechanics there are some  subtleties, and there is no consensus  on the dilemma ``compact vs. extended phase''. \n\nThe proponents of  the suggestion that only the compact phase must be used in the quantum theory of the JJ argue that it is natural to expect that that the states with the phases $\\varphi$ and $\\varphi+2\\pi$ are the same state and  only states with wave functions periodic in  $\\varphi$ with the period $2\\pi$ are possible. This means that the variable canonically conjugated to the phase (moment in a quantum rotator, or charge in a JJ) is quantized. The proponents of the extended  phase argue, that the ``natural expectation'' of the identity of the states with the phases $\\varphi$ and $\\varphi+2\\pi$ is not so natural and is invalid in the case of the JJ because of its interaction with the environment.  Then different phases in the whole interval  $-\\infty<\\varphi <\\infty$ always correspond to different states. This was called ``decompactification of the phase''.  \n\nIt  is important to stress, however, that the compact phase is sufficient for description of states, but not for description of dynamical processes of transitions between states with different phases. In these processes it is necessary  to know  not only the variation of the compact phase but also an integer number $s$ of full $2\\pi$ winding during the process.  It is more convenient instead of two variables to deal only with one variable, which is an extended phase determined in the interval $(-\\infty,\\infty)$\n\\begin{equation}\n\\varphi(t)  =2\\pi s(t) +  \\varphi_c(t).\n      \\ee{comp}\nHere $\\varphi_c$ is the compact phase determined in any interval of the length $2\\pi$. The voltage drop over the JJ is determined by the time derivative of the extended but not the compact phase. The time derivative of the compact phase has unphysical jumps when the phase reaches borders of the $2\\pi$ interval chosen for the compact phase.\nThus, one should not interpret  the requirement of using only the compact phase (phase compactification) literally but interpret it as  the requirement of using only wave functions periodic in the extended phase $\\varphi$ with the period $2\\pi$. Under  the assumption of decompactification this requirement is abandoned.\nThus, the dilemma ``compact vs. extended  phase'' is in fact  the dilemma whether the  states with the phases $\\varphi$ and $\\varphi+2\\pi$  indistinguishable or distinguishable. Nevertheless, further in the paper the dilemma will be called  ``compact vs. extended  phase'' as widely accepted in the literature.\n\n\nActuality of this dilemma  reemerged in the recent dispute about the Dissipative Quantum Phase  Transition (DQPT) \\cite{Sacl,CommMurani,ReplMurani}  between the superconducting and insulating states of a single JJ predicted about 40 years ago \\cite{Schmid,Bulg}.  \\citet{Sacl} claimed that their  experiment and theory proved the absence of  the  DQPT, because the single JJ cannot be an insulator. \n\nThe estimation done in Ref.  \\onlinecite{CommMurani} demonstrated that the experiment of Murani {\\em et al.} was done at the conditions, in which the existing theory did not predict the DQPT. Therefore, the experiment could not provide  any evidence either pro or contra the DQPT. Their theoretical arguments were also rejected, but they deserve an  analysis more detailed than it was possible within  a short Comment \\cite{CommMurani}. In particular, Murani {\\em et al.} \\cite{Sacl,ReplMurani} argued that the conventional theory failed because it  used the extended phase while  only the compact phase must be used. This bring us to the topic of the present paper.\n\nBecause of generality of the aforementioned problems with the phase variable, the paper addresses three systems: quantum rotator, particle in a periodic potential, and single JJ.  In the quantum rotator a particle moves around a 1D ring. The quantum pendulum \\cite{Cond} and an electron moving around a 1D normal ring \\cite{silver,AzbelN,Buttiker,Gefen} are examples of the quantum rotator. The paper explores analogies between these systems, but looking for possible differences at the same time. \n\nIn order to resolve the dilemma ``compact vs. extended  phase'',  it is necessary to answer to three questions:\n\\begin{enumerate}\n\\item\nAre the states with the phases $\\varphi$ and $\\varphi+2\\pi$ indistinguishable in the JJ?\n\\item\nMust the wave function be periodic in $\\varphi$ if the states with the phases $\\varphi$ and $\\varphi+2\\pi$ are indistinguishable?\n\\item\nThe last but not the least: Is it important for the theory of DQPT whether the states with the phases $\\varphi$ and $\\varphi+2\\pi$ are  distinguishable, or not?\n\\end{enumerate}\nThe paper looks for answers to these three questions. \n\n\n\nOur analysis has fully confirmed the final conclusions of the about 40-years old conventional theory of the  DQPT in the small JJ. But it reassessed  justifications of  these conclusions and analogies of the single JJ with other systems  (quantum rotator and particle in a periodic potential).  While in the past it was widely (but not unanimously) believed that the single JJ is an analog of particle in a periodic potential, but not of a mechanical pendulum, we argue that the opposite is true. This means that the states with the phases $\\varphi$ and $\\varphi+2\\pi$ are distinguishable both in the JJ and the quantum rotator.  However, whatever analogy is more correct, the  DQPT must take place both in the JJ and  in the quantum rotator. \n\n\n\\section{The phase in the classical theory} \\label{cl}\n\n\\subsection{Plane rotator in the conjugate variable ``angle--moment''}            \\label{QRc}\n\nThe Hamiltonian of the classical plane rotator  is \n\\begin{equation}\nH={m^2\\over 2 J}+G (1-\\cos \\varphi )-\\varphi N,\n      \\ee{Ho}\nwhere $m$ is the moment of the particle moving in the rotator, $J$ is  the moment of  inertia, $N$ is the external torque, and the periodic in $\\varphi$ potential $\\propto G$\nemerges from a constant force acting on the rotator (the gravity force in the case of the pendulum or the constant  electric field in the case of a charged particle  in a 1D ring). The $\\varphi$-dependent part of the Hamiltonian is the well known washboard potential.\nThe Hamilton equations are\n\\begin{eqnarray}\n{dm\\over dt}= -{\\delta H\\over \\delta \\varphi }=- G\\sin\\varphi +N,\n \\nonumber \\\\\n {d\\varphi \\over dt}={\\delta H\\over \\delta m}={m\\over J}.\n   \\eem{phit}\n\n\nThe Hamiltonian \\eq{Ho} is not periodic in $\\varphi$. This looks as a flaw, since violates the principle that the states with the phases $\\varphi$  and $\\varphi+2\\pi $ are indistinguishable. \nAccording to this Hamiltonian, the energies of these states differ by the energy $2\\pi N$ pumped to the rotator from the environment after full $2\\pi$ winding of the rotator. \n\n\nThe flaw can be eliminated by using another Hamiltonian \n\\begin{equation}\nH={( M+M_0)^2\\over 2 J}+G (1-\\cos \\varphi ), \n            \\ee{Hm}\nwhich is periodic  in $\\varphi$. Here  the moment $M_0$ transferred to the rotator by  the external  torque,\n\\begin{equation}\n{dM_0\\over dt}=N,\n    \\ee{}\nwas introduced. Since $M_0$ emerges from the interaction with the environment, we shall call it {\\em external moment}. \nThe Hamilton equations for the Hamiltonian \\eq{Hm} are\n\n\\begin{eqnarray}\n{d M\\over dt}= -{\\delta H\\over \\delta \\varphi }=- G\\sin\\varphi ,\n\\nonumber \\\\\n{d\\varphi \\over dt}={\\delta H\\over   \\delta  M}={ M+M_0\\over J}.\n   \\eem{phgP}\nThe moment  \n\\begin{equation}\n M ={\\partial {\\cal L}\\over \\partial \\dot \\varphi}\n    \\ee{}\nis the canonical moment determined by the Lagrangian \n\\begin{equation}\n{\\cal L}={(J \\dot \\varphi -M_0)^2\\over 2 J}-G (1-\\cos \\varphi ).\n    \\ee{}\nHowever, the angular  velocity $\\omega=\\dot \\varphi$ of the phase rotation is determined not by the canonical  but by the kinetic moment $m=  M+M_0$. The equations of motion \\eq{phit}  in  the terms of the observables $m$ and $\\varphi$ do not depend on the choice of  the Hamiltonians \\eq{Ho}     or \\eq{Hm}, since they differ by the full time derivative from $\\varphi M_0$, which does not affect the equations of motion. Later on we shall call the gauges with the periodic Hamiltonian like in \\eq{Hm} and with the non-periodic Hamiltonian like in \\eq{Ho} {\\em periodic }and {\\em non-periodic gauge}, respectively.  \n\n\n\nThe terms  {\\em canonical moment} and {\\em kinetic moment} were introduced by the analogy with the {\\em canonical momentum} and {\\em kinetic momentum}  of a charged particle in the electromagnetic field.  Splitting of the kinetic moment $m$ onto the canonical and the external moment obtained  from the environment is purely formal in the classical theory. But this splitting is more important in the quantum theory. \n\n In the  absence of the torque   $ N$, which pumps the moment and the energy to the system, there are two types of motion: (i) an oscillation around the ground state $\\varphi=0$ with $m$ vanishing  in average, and (ii) a monotonic rotation with $\\langle m \\rangle \\neq 0$  $\\varphi(t)$ being a periodic function in the time interval from $-\\infty$ to $\\infty$. \nThe stationary state with constant \n\\begin{equation}\n\\varphi =\\arcsin(N\/G)+2\\pi s\n     \\ee{}\n  is possible if $N <G$.   In this state the kinetic moment $m$ vanishes.  The  states of the rotator at the phases $\\varphi$ and $\\varphi+2\\pi s$ are indistinguishable.  The stationary state with   constant $\\varphi $  at  $N <G$ is an analog of the superconducting state of the JJ, while the regime of monotonic rotation  is an analog of the insulating state of the JJ.  At  $N >G$  the torque  drives the  quantum rotator to rotate with acceleration, but the monotonic  rotation with the angular velocity $\\omega=d\\varphi \/dt$ periodically oscillating around the constant average angular velocity $\\bar \\omega$  is possible  in the presence of  friction.\n\n\n\n\\subsection{Plane rotator vs. particle in a periodic potential}\n\nLet us  compare the rotator dynamics  with the dynamics of a particle with charge $q$ moving in a periodic potential and a classical electromagnetic field. In the latter case the Hamiltonian is \n\n\\begin{equation}\nH={( P- {q\\over c} A)^2\\over 2 m_0 }+G\\left(1-\\cos {2\\pi x\\over l} \\right) +q\\Phi. \n            \\ee{Hg}\nHere $m_0$ is the particle mass, $c$ is the speed of light, $P$ and $A$ are  the $x$-components of the canonical momentum $\\bm P$ and  of the electromagnetic vector potential $\\bm A$ (we consider the 1D problem), and $\\Phi$ is the electromagnetic scalar potential. The gauge transformation,\n\\begin{equation}\nA=A' +\\nabla \\chi(x,t),~~\\Phi=\\Phi' -\\dot   \\chi(x,t),\n     \\ee{GT}\nwith $\\chi(x,t)$ being an arbitrary function of $x$ and $t$,\nyields the Hamiltonian \\eq{Hg} with $A'$ and $\\Phi'$ instead of $A$ and $\\Phi$ and with the full time derivative of $q\\dot   \\chi(x,t)$ added, which  does not affect the Hamilton equations of motion.\n\nThe torque on the plane rotator can be a result  of the magnetic field normal to the plane of the rotator. The Hamiltonian   \\eq{Hg} describes also the dynamics of the plane rotator with $x$ being the coordinate along the circumference of the 1D ring of the rotator and $l$ being the length of this circumference.  The relations between variables in two presentations are \n\\begin{equation}\n\\varphi={2\\pi x\\over l} ,~~M={Pl\\over 2\\pi},~~J={m_0l^2\\over 4\\pi^2} .\n    \\ee{cord}\nThe periodic gauge in Sec.~\\ref{QRc} corresponds to the gauge without the scalar potential $\\Phi$ and with the Hamiltonian \n\\begin{equation}\nH={( P- {q\\over c} A)^2\\over 2 m_0 }+G\\left(1-\\cos {2\\pi x\\over l} \\right), \n            \\ee{HmS}\nThe external moment and the external torque of Sec.~\\ref{QRc} are\n\\begin{equation}\nM_0=-{ql \\over 2\\pi c }A=-\\hbar{\\phi\\over \\phi_0},~~N=\\dot M_0={ ql  \\over 2\\pi}E,\n    \\ee{cor}\nwhere $E=-\\dot A\/c$ is the azimuthal component of the electric field and $\\phi_0=hc\/q$ is the magnetic flux quantum  for the particle of charge  $q$.\nThe external  moment $M_0$  is determined by the  magnetic flux $\\phi=Al$ through the area restricted by the 1D ring of the rotator. The magnetic field is supposed to be  axisymmetric.\n\nThe transformation with the gradient  $\\nabla \\chi = A(t)$ independent from $x$ yields the non-periodic gauge, in which the potential $A(t)$ is absent, but instead the linear in $x$ scalar potential $\\Phi(t)=-  E(t) x=\\dot A(t) x\/c$ appears: \n\\begin{equation}\nH={ P^2\\over 2 m_0 }+G \\left(1-\\cos {2\\pi x\\over l} \\right)- q E x. \n            \\ee{WS}\n \n\nThe scalar potential in the non-periodic gauge is multivalued. \nThis does not produce any problem, since only fields but not potentials are observable quantities.\n\n\n\n Whatever gauge one uses, there is no difference between  dynamics of the rotator and the charged particle in a periodic potential. The dynamics does not depend on whether  the positions of the particle with the coordinates $x$ and  $x+l$ (or angles $\\varphi$ and $\\varphi+ 2\\pi $) are distinguishable,  or not.  Thus, in the classical theory the dilemma ``compact vs. extended  phase'' does not exist and there is no difference between the dynamics in a 1D ring and in the infinite 1D space with a periodic potential.\n\n\n\\section{The phase in the quantum  theory}\n\n\n\\subsection{Axisymmetric quantum  rotator: commutation and uncertainty relations, and wave packets} \\label{WP}\n\n\nThe standard way to go from the classical to the quantum theory is to replace  in the Hamiltonian the canonical moment $ M$  by the operator\n\\begin{equation}\n\\hat { M} =-i\\hbar {\\partial  \\over \\partial \\varphi}.\n    \\ee{Mphi}\n\nGeneral problems  with the canonically conjugate  pair  ``angle  (phase)--moment'' can be discussed for\nthe simple case of  the axisymmetric rotator ($G=0$). This case has already been investigated in the works on persistent currents in 1D normal rings \\cite{Buttiker,Gefen}. For a while we also ignore \n the external moment $M_0$. This means that we ignore any interaction with the environment, either at the present moment, or in the past. \n \n The objection to commutation relation \\eq{ComS} was based on the following calculation of the matrix elements of the commutation relation between two eigenstates of the moment operators with eigenvalues $M_s$ and $M_{s'}$ \\cite{judge,PhasRev,Pegg}:\n\\begin{eqnarray}\n\\int\\limits_0^{2\\pi}\\psi^*_{s'} [\\hat \\varphi,\\hat M]\\psi_s\\,d\\varphi\n\\nonumber \\\\\n=-i\\hbar \\int\\limits_0^{2\\pi}\\psi^*_{s'}\\left[ \\varphi{\\partial \\psi_s\\over \\partial \\varphi}-{\\partial (\\varphi \\psi_s)\\over \\partial \\varphi} \\right]\\,d\\varphi\n\\nonumber \\\\\n=-i\\hbar\\int\\limits_0^{2\\pi}\\left(\\psi^*_{s'} \\varphi{\\partial \\psi_s \\over \\partial \\varphi}+{\\partial \\psi^*_{s'} \\over \\partial \\varphi}\\varphi \\psi_s\\right)\\,d\\varphi\n \\nonumber \\\\\n=\\left. i\\hbar \\psi^*_{s'} \\varphi\\psi_s  \\right|_0^{2\\pi}+(M_s-M_{s'})\\int\\limits_0^{2\\pi}\\psi^*_{s'} \\varphi\\psi_s \\,d\\varphi\n    \\eem{ComC}\nThe opponents of the commutation relation \\eq{ComS} neglected the first term emerging from the borders of the integration interval $(0,2\\pi)$.  Then the diagonal matrix elements ($M_s=M_{s'}$) of the commutator  vanish, while the diagonal matrix elements of the righthand side of the commutation relation \\eq{ComS} are not zero. The justification\nfor ignoring of the border contribution  was that it should not appear in a matrix element of a Hermitian  operator when the integrand is a periodic function of $\\varphi$. \nThe operator  $\\hat \\varphi$ in the commutation relation is not Hermitian and breaks periodicity. \n\nVarious modifications of the commutation relation were suggested (one of them is discussed below).  However, there is another resolution of the problem, which rehabilitates the commutation relation \\eq{ComS}. The border  term in \\eq{ComC}  appears after the integration by parts of only one from two terms in the original commutator. While any of two terms in the commutator separately is non-Hermitian and breaks periodicity, their sum is  Hermitian and does not break periodicity. The matrix element of the commutator can be calculated without integration by parts:\n\\begin{eqnarray}\n\\int\\limits_0^{2\\pi}\\psi^*_{s'} [\\hat \\varphi,\\hat M]\\psi_s\\,d\\varphi\n\\nonumber \\\\\n=-i\\hbar \\int\\limits_0^{2\\pi}\\psi^*_{s'}\\left( \\varphi{\\partial \\psi_s\\over \\partial \\varphi}-\\varphi {\\partial \\psi_s\\over \\partial \\varphi} -\\psi_s\\right)\\,d\\varphi\n\\nonumber \\\\\n=i\\hbar \\int\\limits_0^{2\\pi}\\psi^*_{s'}\\psi_s\\,d\\varphi.\n    \\eem{ComP}\nThis is equal to the matrix element of the righthand side of the commutation relation. Similar arguments rehabilitating the standard commutation relation \\eq{ComS} were presented by \\citet{LossM}.\n\nWe checked the commutation relation using the wave functions in the continuous space of the phase $\\varphi$. Another route is to do it in the discrete space of quantized moments $M$. Then one encounters the  problem that because of discreteness of $M$ the expression conjugate to \\eq{Mphi}\n\\begin{equation}\n\\hat \\varphi =i\\hbar {\\partial  \\over \\partial M}\n    \\ee{phiM}\nfor the operator $\\hat \\varphi$ in the moment space is invalid. Instead one can use the operator $e^{i\\hat \\varphi}$, which shifts from one quantized eigenvalue of the operator $\\hat M$   to the next  one.\nThe commutation relation with this operator is\n\\begin{equation}\n[e^{i\\hat \\varphi},\\hat M]=-\\hbar e^{i\\hat \\varphi}.\n    \\ee{eph}\n\nThe operator $e^{i\\hat \\varphi}$ is a superposition of two Hermitian operators $\\cos \\hat \\varphi$ and $\\sin \\hat \\varphi$. The commutation relations for these operators, which are equivalent  to \\eq{eph}, were suggested by \\citet{suss}. Although the commutation relation \\eq{eph} contains only Hermitian  operators, its expansion in $\\hat \\varphi$ consists of non-Hermitian  operators, which must be treated correspondingly. A failure of some operation valid only for Hermitian  operators, means the failure of the operation, but not of the commutation relation.\n\nThe problem with the canonical commutation relation naturally leads to the problem with the uncertainty relation \\eq{unc}. The uncertainty relation is derived from the analysis of semiclassical wave packets, which demonstrates the correspondence principle: the transition from the quantum mechanical to the classical description. The wave packet is formed by a superposition of the states with  moments $M$ in the interval of the width $\\Delta M$. In the continuous moment space the superposition is determined by an integral. In the discrete moment space the integral must be replaced by a sum over the quantized values of the moment.\n \n \n \n Let us assume that the phase uncertainty is essentially less than $2\\pi$.  Then the moment uncertainty\n\\begin{equation}\n\\Delta M \\gg {\\hbar\\over 4\\pi} \n   \\ee{}\nis much larger than the distance between quantized values of the moment. Then the quantization can be ignored, and the summation determining the wave packet can be replaced by the integration. This provides a sufficiently accurate description of the wave packet within the phase interval $2\\pi$. \n\nBut there is a hurdle in this picture. The original wave function of the wave packet with summation over quantized values of the moment is periodic in $\\varphi $ because any term of the sum is periodic. However, the replacing of summation by integration definitely breaks the periodicity. The flaw is easily healed. The wave packet is replaced by the periodic chain of packets. The procedure can be considered as a compactification of phase, which the proponents of the compact phase insist on. Namely, one calculates the phase only in one $2\\pi$ interval and then continues this wave function periodically on all other phase intervals.  At the same time this demonstrates that the compactification is necessary only due to inaccuracy  of the approximation and is not needed for a sufficiently exact analysis. \n\n\nAnother situation emerges if one tries to construct a wave packet with the phase uncertainty $ \\Delta \\varphi$ exceeding $2\\pi$.  In this case the moment uncertainty $\\Delta M <\\hbar\/4\\pi$ is less than the moment quantum, the summation reduces to only one term, and the picture of wave packets fails. Then the physical meaning of the uncertainty relation \\eq{unc} becomes unclear.\n\nThere were suggestions to modify the standard uncertainty relation \\eq{unc} as a reaction to the aforementioned problem \\cite{judge,Pegg}. They were based on the concept of the compact phase, which assumed that the phase uncertainty cannot exceed $2\\pi$. This concept ignores that a phase fluctuation cannot be described only by the fluctuation of the compact phase (Sec.~\\ref{cl}) . A number $s$ of full $2\\pi$ rotations [see \\eq{comp}] in the course of the fluctuation is also important. We do not dwell more on this issue, since our analysis of the slow dynamics is based on the adiabatic approximation and does not use the wave packet concept. \n\n\n\\subsection{Particle moving in a 1D ring vs. particle moving in an infinite 1D space} \\label{per}\n\nLet us now compare the quantum mechanical dynamics of  a particle moving  in a 1D ring of the rotator and a  particle moving in the infinite 1D space. As in the previous subsection,\nwe ignore the periodic potential $G\\left(1-\\cos {2\\pi x\\over l} \\right)$. This allows to deal with simple analytical solutions of the Schr\\\"odinger equation.\n\nIn the quantum mechanics the canonical momentum becomes an operator:\n\\begin{equation}\n\\hat P=-i\\hbar {\\partial \\over \\partial x}.\n     \\ee{}\nThe quantum mechanical version of the Hamiltonian  \\eq{Hg} and the Schr\\\"odinger equation at $G=0$ in the periodic gauge  are \n\\begin{equation}\nH={1\\over 2 m_0}\\left|-i \\hbar{\\partial \\psi \\over \\partial x} -{q\\over c}A(t)\\psi\\right|^2, \n            \\ee{Hcx}\n\\begin{equation}\ni\\hbar {\\partial \\psi \\over \\partial t}=-{1\\over 2m_0}\\left[ \\hbar{\\partial  \\over \\partial x} -i{q\\over c}A(t)\\right]^2\\psi  .\n            \\ee{Shcx}\nThe Schr\\\"odinger equation  has a solution for an arbitrary time dependence of  $A(t)$:\n\\begin{equation}\n\\psi(\\varphi,t) = e^{iP x \/\\hbar-i \\int^t{{\\cal E}(t')\\over \\hbar}dt'},\n    \\ee{wfc}\nwhich is an eigenstate on the canonical momentum with the  eigenvalue $P$.  Here the time dependent energy is\n\\begin{equation}\n{\\cal E}(t)={[P -{q\\over c}A(t)]^2\\over 2m_0}.\n     \\ee{}\nThe particle  velocity \n\\begin{equation}\nv ={dx\\over dt} ={d{\\cal E}\\over d P}={P -{q\\over c}A(t)\\over m_0}\n    \\ee{omega0}\ndepends on the kinetic momentum $p=P -{q\\over c}A(t)$ and is well-defined, while the coordinate $x$ itself is not defined at all. There is an equal probability for any value of $x$.  \nAn electric field  $E=-\\dot A\/c$ monotonically accelerates the particle, as in the classical theory.\n\nIn a constant electric field\n\\begin{equation}\n\\psi(\\varphi,t) = e^{iP x \/\\hbar- {i(P+qEt)^3\\over 6m_0 qE}}.\n    \\ee{}\n\n\n\nIn the quantum mechanics  the difference of the particle dynamics in the 1D infinite space  and in the 1D ring  becomes  important. In the former case any value of $P$ is allowed. In the latter case  \nthe canonical momentum $P$ is quantized and cannot differ from the values $sh\/l$ with integer $s$. Only at these quantized  values the wave function \\eq{wfc} is periodic with the period $l$.  \nIn the variables ``moment--angle'' the quantized values of the canonical moment $M=Pl\/2\\pi$ [see \\eq{cord}] are $s\\hbar$.\nThe plot the energy vs. the external moment $M_0$ for different quantized values of the canonical moment  is shown in Fig.~\\ref{f1}(a).\n\n \n\\begin{figure}[!b]\n\\includegraphics[width=0.4 \\textwidth]{Energ} \n \\caption{ Plot the energy vs. the external moment $M_0$ at various quantized values of the canonical moment $M=s\\hbar$. (a) Axisymmetric rotator.  (b) Quantum rotator in a constant field.      \\label{f1}}\n \\end{figure}\n\n\n\n\nIn the quantum theory the gauge transformation \\eq{GT} must be accompanied by the transformation of the wave function \\cite{LLqu}:\n\\begin{equation} \n\\psi =\\psi'e^{iq\\chi\/\\hbar c}.\n      \\ee{GTp}\nAfter the transformation with $\\chi =xA=-xc\\int^tE(t')dt'$,\n\\begin{equation} \n\\psi =\\psi'e^{i qAx\/\\hbar c},\n      \\ee{gauge}\n  from the periodic to the non-periodic gauge, the Hamiltonian and the  Schr\\\"odinger equation become\n\\begin{equation}\nH={\\hbar^2\\over 2 m_0}\\left|{\\partial \\psi' \\over \\partial x}\\right|^2- qE(t)x|\\psi'|^2,\n            \\ee{Hwb0}\n\\begin{equation}\ni\\hbar {\\partial   \\psi' \\over \\partial t}=-{\\hbar^2\\over 2m_0}{\\partial^2 \\psi '\\over \\partial x^2} - qE(t)x\\psi'.\n   \\ee{Sht0}\nThe gauge transformation  \\eq{gauge} yields an non-stationary state with the non-periodic wave function\n \\begin{eqnarray}\n\\psi'=\\psi e^{-qAx\/c} = e^{i\\left(P +qE t\\right) x \/\\hbar-i \\int^t{{\\cal E}(t')\\over \\hbar}dt'}.\n    \\eem{nonpf}\n In a constant electric field\n\\begin{equation}\n\\psi'(\\varphi,t) = e^{i(P+qEt) x \/\\hbar- {i(P+qEt)^3\\over 6m_0 qE}}.\n    \\ee{}\n\n\n\nAfter the gauge transformation the canonical and the  kinetic momentum do not differ and are determined by the operator\n\\begin{equation}\n\\hat p'=\\hat P'=-{\\partial \\over \\partial x}\n    \\ee{}   \nin the space of functions $\\psi'$. In the quantum rotator  the quantization of the canonical momentum must be done in the periodic but not in the non-periodic gauge. This means that not the momentum $p'$ but the canonical momentum $P=sh\/l$  is equal  to an integer number $s$ of the momentum quanta. \n\nIn the non-periodic gauge the wave function is non-periodic since the gauge transformation \\eq{GTp} and the Hamiltonian \\eq{Hwb0} are non-periodic. One can rewrite  \\eq{nonpf} as \n\\begin{eqnarray}\n\\psi'= e^{iPx \/\\hbar-i \\int^t{{\\cal E}'(t')\\over \\hbar}dt'},\n    \\eem{}\nwhere \n\\begin{equation}\n{\\cal E}'(t)={\\left(P +qEt\\right)^2\\over 2J}-qEx\n     \\ee{}\nis the energy after the  gauge transformation. It is evident that the  wave function is not periodic because of the non-periodic term $-qEx$ in the energy.  It is impossible to satisfy the requirement of the wave function periodicity in the non-periodic gauge. If the wave function  is periodic at some moment of time it will become non-periodic at the next moment of time because of the non-periodic Schr\\\"odinger equation.\n\n\n\nThe loss of periodicity of the wave function in the non-periodic gauge should not be a matter of concern, as well as not  a matter of concern is the non-periodic electric scalar potential.  The phase factor, which makes the wave functions $\\psi(x)$ and $\\psi(x+l)$ different, means that the state is described by a multivalued wave function. The property of the gauge transformation to make the wave  function multivalued was pointed out by \\citet{LLqu} in Sec. 111 of their book. Multivaluedness (non-periodicity) of the wave function of the quantum rotator compensates multivaluedness (non-periodicity) of the washboard potential  in the non-periodic gauge \\cite{david}.\n\n\nIn summary, the important and the only difference between the dynamics of the particle in the quantum rotator and the particle moving in the infinite 1D space is that the Hilbert space of wave functions in the former case is discrete and is a subspace of the continuous Hilbert state in the latter case.\n\n\\subsection{Quantum  rotator in an external constant field} \\label{rcf}\n\nWhile in Sec.~\\ref{per} the variables ``coordinate--momentum'' were more convenient for comparison of the rotator with the particle moving in the infinite 1D space, here we return back to the variables ``angle (phase)--moment'', which are more convenient for comparison with the JJ.\n\nAt the presence of the external constant field the quantum mechanical version of  the  Hamiltonian \\eq{Hm} in the periodic gauge is\n\\begin{equation}\nH={1\\over 2 J}\\left|-i \\hbar{\\partial \\psi \\over \\partial \\varphi} +M_0\\psi\\right|^2+G (1-\\cos \\varphi ) |\\psi|^2.\n            \\ee{Hms}\nThe Schr\\\"odinger equation for this Hamiltonian is\n\\begin{equation}\ni\\hbar {\\partial \\psi \\over \\partial t}=-{1\\over 2J}\\left(\\hbar{\\partial \\over \\partial \\varphi}+iM_0\\right)^2\\psi +G (1-\\cos \\varphi ) \\psi.\n   \\ee{Sh}\n\n\nAccording to  the Bloch theorem,  any stationary  solution of \\eq{Sh} is the Bloch function\n\\begin{equation}\n\\psi(\\varphi,t)=u(\\tilde M+M_0,\\varphi)e^{i (\\tilde M \\varphi- E_0 t)\/\\hbar},\n   \\ee{BF}\nwhere $u(\\tilde M+M_0,\\varphi)=u(\\tilde M+M_0,\\varphi+2\\pi)$ is a periodic in  $\\varphi$  function and   $\\tilde M$ is a canonical quasimoment (analog of the canonical quasimomentum in the solid body theory \\cite{LLstPh2}). The energy spectrum of Bloch states consists of bands  with forbidden  gaps between them. In the quantum rotator the  Bloch wave function must be periodic in $\\varphi$. It is possible only  for quantized values  of the canonical quasimoment  $\\tilde M=s\\hbar$. \n\n\n\nWe consider only the lowest band with the energy $E_0(\\tilde M  +M_0)$, which depends on  the kinetic quasimoment  $\\tilde m=\\tilde M  +M_0$. \nBy the analogy with the quasimomentum and the coordinate operators for a particle in a periodic potential, one can consider the energy $E_0(\\tilde M  +M_0)$ as a Hamiltonian   \\cite{LLstPh2}, \nwhich yields the Hamilton equations \n\\begin{equation}\n{d \\tilde M\\over dt}=- {\\partial E_0\\over \\partial \\varphi}=0,\n   \\ee{Mt}\n\\begin{equation}\n\\omega={d \\varphi\\over dt}={\\partial E_0(\\tilde M  +M_0)\\over \\partial \\tilde M}.\n            \\ee{omega}\nWhile  the canonical quasimoment $\\tilde M$ does not vary in time,  the kinetic quasimoment $\\tilde m=\\tilde M  +M_0$ depends on time:\n\\begin{equation} \n{d\\tilde m\\over dt}={d\\tilde M\\over dt}  +\\dot M_0=\\dot M_0.\n      \\ee{tlM}\nIn general, these equations must be operator equations for the conjugate operators of the canonical quasimoment   $\\tilde M$ and of the angle $\\varphi$  \\cite{LLstPh2}. But if the torque is weak, one can use the adiabatic approximation with $M_0$ being a slowly varying adiabatic parameter. This allows to assume that at any moment the state does not differ essentially from the eigenstate of  the canonical quasimoment with the eigenvalue $\\tilde M$  at fixed $M_0$. Then Eqs.~(\\ref{Mt}--\\ref{tlM}) can be treated as classical equations. \n\n \n In the solid body theory the classical treatment of Eqs.~(\\ref{Mt}) and  (\\ref{omega}) is sometimes justified by considering them as written for  semiclassical wave packets \\cite{ziman}. Since for the quantum rotator the concept of wave packets is problematic (see Sec.~\\ref{WP}), it is important that for this  justification we used the adiabatic principle, but not the concept of wave packets.\n\n\nThe function $E_0(\\tilde m)$ is determined by the solution of the Schr\\\"odinger equation in Mathieu functions. Close to the bottom of the band\n\\begin{equation}\n E_0(\\tilde M  +M_0)={(\\tilde M  +M_0)^2\\over 2 J^*},~~\\omega={\\tilde M  +M_0\\over J^*},\n     \\ee{}\nwhere \n\\begin{equation}\nJ^* = \\left[{\\partial ^2E_0(\\tilde M  +M_0)\\over \\partial \\tilde M^2}\\right]^{-1}\n    \\ee{}\nis  the effective moment of inertia, an analog of the effective mass in the Bloch theory for solids.\n\n\n\n\n\nIn the weak binding limit $G \\ll \\hbar^2\/J$ the energy in the Brillouin zone $-\\hbar\/2 < \\tilde m < \\hbar\/2$ is $E_0={\\tilde m^2\\over 2J}$ excepting the close vicinity to the zone borders.\nThe effective moment of inertia $J^*$ does not differ from the bare moment of inertia $J$. The dependence of the energy on the external moment $M_0$  in the weak binding limit is shown for two Bloch bands at quantized values of $\\tilde M=s\\hbar$ in Fig.~\\ref{f1}(b).\n\nIn the strong binding limit $G \\gg \\hbar^2\/J$ there is the narrow band\n \\begin{equation}\n   E_0=\\Delta \\left(1-\\cos {2\\pi \\tilde m\\over \\hbar}\\right), \n      \\ee{}\nwhere $\\Delta $ is the band half-width, which goes to zero at $GJ\/\\hbar^2\\to \\infty$. The effective  moment of inertia is\n\\begin{equation}\nJ^*={\\hbar^2 \\over 4\\pi^2\\Delta}.\n   \\ee{}\n\nThe band energy has extrema at $M_0=s\\hbar$, where the phase angular velocity $\\omega$ vanishes. Thus, at zero external moment $M_0$, i.e., in the absence of any connection with the environment, either at the present moment, or in the past, the monotonic phase rotation is impossible. This follows  from the analysis of the quantum pendulum  \n\\cite{Cond} and of the quantum rotator in a constant electric field \\cite{silver} ignoring the connection  with the environment. \n \n Impossibility of monotonic phase rotation without any connection with the environment can be explained by the following arguments. In  an axisymmetric rotator, i.e., without an external constant field, there are two degenerate eigenstates  of fixed energy, which are either two states $e^{\\pm i M\\varphi\/\\hbar}$ with rotating phase, or states $\\cos {M\\varphi\\over \\hbar}$ and  $\\sin {M\\varphi\\over \\hbar}$  with  vanishing average angular velocity $\\langle\\dot \\varphi\\rangle$. But in a coherent superposition of  degenerate states $\\cos {M\\varphi\\over \\hbar}$ and  $\\sin {M\\varphi\\over \\hbar}$ the  nonzero angular velocity $\\langle\\dot \\varphi\\rangle$ is possible. However, whatever weak phase dependent potential (even a single impurity)  breaks the axial symmetry  and lifts degeneracy of states $\\cos {M\\varphi\\over \\hbar}$ and  $\\sin {M\\varphi\\over \\hbar}$. Then the superposition of two states  is not an eigenstate of the energy operator. In any eigenstate of the energy phase rotation is impossible.\n \n\nLet us apply  a constant weak torque $N$ to the rotator.  The external moment $M_0=Nt$  is proportional to time, and the time  can be excluded from Eqs.~(\\ref{omega}) and (\\ref{tlM}). This yields the equation \n\\begin{equation}\nN{d \\varphi\\over d\\tilde m}={\\partial E_0(\\tilde m)\\over \\partial \\tilde m}.\n            \\ee{Tm}\nThe equation describes Bloch oscillations with the  time period\n \\begin{equation}\n T={\\hbar \\over N }. \n       \\ee{t}\nWhile in the absence of the external field ($G=0$), the torque produces rotation with acceleration as in the classical theory (Sec.~\\ref{per}), in the presence of the periodic external field\nthe group velocity  $\\partial E_0(\\tilde m)\/\\partial \\tilde m$ is a periodic function of $\\tilde m$, and  the phase  angular velocity $\\omega$ performs periodic Bloch oscillations.  The angular velocity   vanishes after averaging over the time  and the amplitude of the phase oscillation is  determined by the  width $\\Delta E_0=E_{0\\mbox{max}}-E_{0\\mbox{min}}$ of the Bloch band:  \n\\begin{equation}\n\\Delta \\varphi ={ \\Delta E_0\\over N }.\n     \\ee{}\nThus, any finite torque makes  a monotonic rotation impossible. However, in the limit $N \\to 0$ when the period  $ T$ becomes much longer than the time of observation, one cannot discern the Bloch oscillation from monotonic rotation.\n\nNext we  phenomenologically introduce dissipation. The environment not  only pumps the moment into the rotator, but also provides a friction torque proportional to the phase angular velocity:\n\\begin{equation}\n\\dot M_0= N -f  \\omega = N -f {\\partial E_0(\\tilde m)\\over \\partial \\tilde m}.\n        \\ee{dis}\nHere $f$ is the friction coefficient. This equation has a solution with constant   $M_0$ and  the phase angular velocity \n\\begin{equation}\n\\omega={N\\over f}.\n     \\ee{Tf}\nNote that  no moment is pumped to the rotator since the moment $M_0$ does not vary in time. The external torque $ N$ is balanced by the friction torque $f\\omega$, i.e., the pumped moment  is returned back to the environment.\n\nRotation of the particle of charge $q$ with the angular velocity $\\omega$ produces the current $j=q\\omega\/2\\pi$. At the same time,  the torque is connected with the electric field $E$ [see \\eq{cor}]. Thus, \\eq{Tf} is in fact the Ohm law $j=El \/R_r$, where \n\\begin{equation}\nR_r ={4\\pi^2 f\\over q^2}\n     \\ee{Rf}\nis the resistance to the circular current $j$ around the 1D ring of the quantum rotator.\n\n\\begin{figure}[!t]\n\\includegraphics[width=0.4 \\textwidth]{VIcurve} \n \\caption{Plot the average angular velocity $\\bar \\omega$ vs. torque $N$ for the quantum rotator and plot the average voltage $\\bar V$ vs. current $I$ for the JJ in dimensionless variables.  Curve 1:  Weak binding limit, $\\omega_0=2\\hbar\/J$, $N_0=\\hbar f\/2 J$, $V_0=e\/C$, $I_0=e\/RC$. Curve 2: Strong binding limit, $\\omega_0=\\hbar\/2\\pi J^*$, $N_0=\\hbar f\/2\\pi  J^*$, $V_0=e\/\\pi C^*$, $I_0=e\/\\pi RC$.       \\label{f2}}\n \\end{figure}\n\n\n\nThe derivative $\\partial E_0(\\tilde m)\/ \\partial \\tilde m$ of the periodic function has a maximum which determines the  maximum of the phase angular velocity $\\omega_m$. When the torque $N$ becomes larger than $f  \\omega_m$ the steady phase rotation is  impossible and the Bloch oscillation starts. In the presence of the dissipation Eqs.~(\\ref{Tm}) and (\\ref{t}) become\n\\begin{equation}\n{d \\varphi\\over d\\tilde m}=\\frac{{\\partial E_0(\\tilde m)\\over \\partial \\tilde m}}{N-f\\ {\\partial E_0(\\tilde m)\\over \\partial \\tilde m}},\n            \\ee{Tmf}\n \\begin{equation}\n T=\\int\\limits_{-\\hbar\/2}^{\\hbar\/2}{d\\tilde m \\over N-f {\\partial E_0(\\tilde m)\\over \\partial \\tilde m }}. \n       \\ee{tf}\nNow the phase not only oscillates but also rotates with the average angular velocity\n\\begin{equation}\n\\bar \\omega ={1\\over T} \\int\\limits_{-\\hbar\/2}^{\\hbar\/2}{{\\partial E_0(\\tilde m)\\over \\partial \\tilde m } \\over N-f {\\partial E_0(\\tilde m)\\over \\partial \\tilde m }}d\\tilde m={N\\over f}-{\\hbar \\over fT}.\n   \\ee{omf}\nAt large $N$ the average angular velocity decreases as $1\/N$. \n\nIn the weak binding limit $G \\ll \\hbar^2\/J$ \n\\begin{equation}\nT= {J\\over f} \\ln \\frac{1+{\\hbar f\\over 2N J}}{1-{\\hbar f\\over 2N J}} ,~~\\bar \\omega={N \\over  f}- {\\hbar  \\over J}\\ln^{-1} \\frac{1+{\\hbar f\\over 2N J}}{1-{\\hbar f\\over 2N J}} .\n     \\ee{WC}\n In the strong binding limit $G \\gg \\hbar^2\/J$\n\\begin{equation}\nT= {J^*\\over f}{1 \\over \\sqrt{{J^{*2}N^2\\over \\hbar^2 f^2 }- {1\\over 4\\pi^2}}},~~\\bar \\omega ={  N\\over f }-{\\hbar \\over J^*}\\sqrt{{J^{*2}N^2\\over \\hbar^2 f^2 }- {1\\over 4\\pi^2}} .\n     \\ee{WCs}\nThe dependences of the average angular velocity $\\bar\\omega$ on the torque $N$ in the weak and the strong limit are shown in the dimensionless variables  in Fig.~\\ref{f2}. \n\nThe gauge transformation \\eq{gauge} in the variables ``phase--moment'' is \n\\begin{equation} \n\\psi =\\psi'e^{-M_0\\varphi\/\\hbar }.\n      \\ee{gaug}\nIt transforms the Hamiltonian \\eq{Hms} and the  Schr\\\"odinger equation \\eq{Sh} to \n\\begin{equation}\nH={\\hbar^2\\over 2 J}\\left|{\\partial \\psi' \\over \\partial \\varphi}\\right|^2+[G (1-\\cos \\varphi )- \\dot M_0 \\varphi]|\\psi'|^2,\n            \\ee{Hwb}\n\\begin{equation}\ni\\hbar {\\partial \\psi' \\over \\partial t}=-{\\hbar^2\\over 2J}{\\partial^2 \\psi' \\over \\partial \\varphi^2}+[G (1-\\cos \\varphi )- \\dot M_0 \\varphi] \\psi',\n   \\ee{Sht}\nwhich are not periodic in $\\varphi$. The  Bloch  wave function after the transformation is also non-periodic:\n\\begin{eqnarray}\n \\psi'(\\varphi)=u(\\tilde M+M_0,\\varphi)e^{i \\left[(\\tilde M+M_0) \\varphi- \\int^{t}E_0( t')dt'\\right]\/\\hbar}\n \\nonumber \\\\\n=u(\\tilde M+M_0,\\varphi)e^{i\\left[\\tilde M\\varphi-\\int^{t}E'_0 (t')dt' \\right]\/\\hbar}\n     \\eem{M0}\nwhere\n\\begin{equation}\nE'_0=E_0(\\tilde M+M_0) -\\dot M_0 \\varphi=E_0(\\tilde M+M_0) -N\\varphi\n     \\ee{}\nis  the energy in the non-periodic gauge.\nFurther discussion of the non-periodic gauge does not differ essentially from the discussion of this gauge for the axisymmetric rotator (Sec.~\\ref{per}). The wave function is not periodic because in the non-periodic gauge the  energy with the washboard potential  is non-periodic.\n\n \n \\begin{figure}[!t]\n\\includegraphics[width=0.4 \\textwidth]{Energ0} \n \\caption{The band energy ${\\cal E}_0$ vs. the external moment $M_0$. (a) The canonical quasimoment $\\tilde M =s\\hbar$ is quantized. (b) The canonical quasimoment $\\tilde M $ is not quantized. There is a continuous manifold of curves for various values of $\\tilde M$.       \\label{f1.1}}\n \\end{figure}\n\n\n\n Let us address now the case of the non-quantized canonical quasimoment $\\tilde M$ (particle in a periodic potential in the infinite 1D space). Figure \\ref{f1.1} compares the dependence of the energy ${\\cal E}_0$ in  the lowest Bloch band on the external moment $M_0$ for the quantized [Fig.~\\ref{f1.1}(a)] and the non-quantized  [Fig.~\\ref{f1.1}(b)] canonical quasimoment $\\tilde M$.  Instead of a single curve for quantized  $\\tilde M$ one has a continuous manifold of curves. Figure~\\ref{f1.1}(b), however, shows only a discrete manifold of curves in order to demonstrate that the curves are obtained from the single curve in Fig.~\\ref{f1.1}(a) by a shift without deformation. \n\n \n Despite this difference between quantized and non-quantized $\\tilde M$, for the regimes discussed above (monotonous phase rotation and Bloch oscillation)  the effect of quantization is practically absent.  The dynamics of these regimes is governed by the kinetic quasimoment $\\tilde m =\\tilde M+M_0$. In the absence of quantization the division of  $\\tilde m$ into  quantized $\\tilde M$ and non-quantized $M_0$ is meaningless since the both are not quantized. If in Fig.~\\ref{f1.1}(b) one plots the energy as a function of $\\tilde m$ instead of $M_0$ this yields the same single curve as in  Fig.~\\ref{f1.1}(a).\n\nSummarizing, the slow dynamics of the particle moving in the 1D ring of the rotator with indistinguishable states with the phases $\\varphi$ and $\\varphi+2\\pi$ does not differ from the dynamics of the particle moving in the infinite 1D space with the periodic potential, when the phases $\\varphi$ and $\\varphi+2\\pi$ correspond to different states.\nThis is because only one Bloch state participates  in the adiabatic processes. During its tuning by the external torque there are neither  intraband,  nor interband\ntransitions between Bloch states.\n\n\n\\section{JJ and DQPT}\n\nThere is one to one correspondence (ideal mapping) between the quantum rotator in the constant external field and the single JJ. The correspondence between variables of two systems is shown in Table 1. \n\n\\begin{table}[ht\n\\begin{tabular}{|l|l|} \\hline  {\\bf Rotator}~~~~~~~~~~~~~~~~ & {\\bf JJ}\n \\\\\n  &   \\\\ \\hline\n Canonical moment $M$ & Canonical charge $Q \\to {2e \\over \\hbar} M$ \\\\ \\hline\n  External moment $M_0$ &  External charge $Q_0 \\to {2e \\over \\hbar}M_0$ \\\\ \\hline \n Torque $N$ &  Electrical current $I\\to {2e \\over \\hbar}N $ \\\\ \\hline\n Rotation angle $\\varphi$ & Quantum mechanical phase $\\varphi$ \\\\ \\hline\nAngular velocity $\\omega=\\dot  \\varphi$ & Voltage  $V= {\\hbar\\over 2e}\\dot \\varphi$ \\\\ \\hline\nMoment of inertia $J$ & Capacitance  $C\\to {4e ^2\\over \\hbar^2}J$ \\\\ \\hline\nFriction coefficient $f$ & Conductance $1\/R\\to {4e ^2\\over \\hbar^2 }f$  \\\\ \\hline\n \\end{tabular}\n\\caption{Correspondence of variables of the rotator and the JJ} \\end{table}\n\n\nAs well as in the case of quantum rotator, the theory of the JJ can use either  the periodic gauge, in which the Hamiltonian and the wave function are periodic in $\\varphi$, or  the non-periodic gauge, in which both the Hamiltonian and the wave function  are not periodic in $\\varphi$. \n\nTranslating the periodic  Hamiltonian \\eq{Hms} and the  Schr\\\"odinger equation \\eq{Sh} for the rotator to the JJ one obtains\n\\begin{equation}\nH={1\\over 2 C}\\left|-i 2e{\\partial \\psi \\over \\partial \\varphi} +Q_0\\psi\\right|^2+E_J (1-\\cos \\varphi ) |\\psi|^2,\n            \\ee{HmQ}\n\\begin{equation}\ni\\hbar {\\partial \\psi \\over \\partial t}=-{1\\over 2C}\\left(2e{\\partial \\over \\partial \\varphi}+iQ_0\\right)^2\\psi +E_J (1-\\cos \\varphi ) \\psi.\n   \\ee{ShQ}\nHere $Q_0$ is the external charge, while the canonical charge is an operator\n\\begin{equation}\n\\hat Q =-2ie{\\partial \\over \\partial \\varphi}.\n    \\ee{}\nThe gauge transformation analogous to \\eq{gauge},\n\\begin{equation}\n\\psi=\\psi' e^{-Q_0 \\varphi \/2e},\n     \\ee{}\nyields the   Hamiltonian and the  Schr\\\"odinger equation  in the non-periodic gauge. The wave function  is periodic in $\\varphi$ in the periodic gauge, but not periodic in the non-periodic gauge. \n\nSolutions of the  Schr\\\"odinger equation \\eq{ShQ} are Bloch functions \n\\begin{equation} \n\\psi(\\varphi,t) =u(\\tilde Q+Q_0)e^{i(\\tilde Q\\varphi\/2e -E_0t\/\\hbar)},\n      \\ee{}\nwhere  $\\tilde Q$ is the canonical quasicharge, which is quantized in the JJ, and $u(\\tilde Q+Q_0)$ is a periodic function of the kinetic quasicharge $\\tilde q= \\tilde Q+Q_0$ with the period $2e$.\n\nThe further analysis is similar to that for the quantum rotator (Sec.~\\ref{rcf}). The analog of  \\eq{dis} is Kirchhoff's law\n\\begin{equation}\n\\dot {\\tilde q}=\\dot  Q_0 =I - {V\\over R},\n   \\ee{}\nwhere \n \\begin{equation}\nV={\\hbar \\over 2e}{d\\varphi\\over dt} \n     \\ee{}\nis the voltage drop across the JJ. The ohmic resistance \n\\begin{equation}\nR= {R_sR_{qp}\\over R_s+R_{qp}},\n   \\ee{}\nis determined by the resistance $R_{qp}$ of the normal channel in the JJ and by the resistance $R_s$ of the external shunt parallel to the JJ.\n\nAt small current $I$ $\\dot Q_0=0$, and the phase rotates with the constant angular velocity, i.e., at the constant voltage $V=IR$. This means that the whole current goes through the ohmic channel, and the JJ is an insulator. The insulating state is possible as far as the voltage $V$ dies not exceeds the voltage $V_0$ equal to the maximum of the derivative ${\\partial E_0\/ \\partial Q_0}$ in the Bloch band. In the limits of weak and strong binding\n\\begin{equation}\nV_0=\\left\\{\\begin{array}{ccc}  {e\\over C} & &   E_J\\ll {e^2\\over C} \\\\ {e\\over\\pi C^*}&  & E_J\\gg {e^2\\over C}   \\end{array}\\right.  .\n   \\ee{}\nThe voltage $V_0$ is  the electric breakdown voltage of the insulator \\cite{CommMurani}. At $V>V_0$  the steady rotation of the phase is impossible. Instead the Bloch oscillation regime takes place accompanied by a slow drift of the phase. The JJ becomes a conductor.  The $VI$ curve of the JJ  is described by the same plot as the plot ``angular velocity vs. torque'' for the quantum rotator shown in Fig.~\\ref{f2}. The $VI$ curves in Fig.~\\ref{f2} were calculated for the JJ by \\citet{widom2} and  by Averin, Likharev, and Zorin \\cite{AverLikh,Likh}. \\citet{widom2} called the voltage maximum on this curve current-voltage anomaly. \n\\citet{Schon} called it Bloch nose. The corresponding maximum on the curve ``resistance--current'' was called the Coulomb blockade bump \\cite{SI,Penttila2001,CommMurani}.\n\nWhile \\citet{widom2}  calculated the $VI$ curve using the analogy with the pendulum, i.e., assuming that the states with the phases $\\varphi$ and $\\varphi+2\\pi$ are  identical, Averin, Likharev, and Zorin \\cite{AverLikh,Likh} assumed that they are not identical as in the case of  a particle in a periodic potential. Nevertheless, the both groups obtained the same $VI$ curves in agreement with our conclusion in the end of Sec.~\\ref{rcf}. \n\nThe possibility of monotonic phase rotation in the Bloch band theory is due to quantum tunneling between neighboring wells of the periodic potential. Dissipation can suppress quantum tunneling \\cite{CL}. Then the particle (virtual particle in the JJ case) becomes localized in one of the wells of the periodic potential. This is the superconducting state of the JJ. The transition from the superconducting to the insulating state is the DQPT. \n\nThe DQPT is a joint effect of Coulomb interaction, dissipation, and quantum mechanics. The Coulomb blockade of Cooper pairs makes the JJ an insulator at small bias. However, it is effective only if the Coulomb energy $E_C= e^2\/C$ exceeds the quantum-mechanical uncertainty $\\hbar \/\\tau$ \\cite{Tin}, where $\\tau=RC$ is the time of the charge relaxation in the circuit.  According to the condition $E_C \\sim \\hbar\/\\tau$, the DQPT is expected at the resistance $R$ of the order of the quantum resistance  $R_q=h\/4e^2$. This qualitative estimation \\cite{Tin,CommMurani} agrees with the more detailed and accurate theory predicting the DQPT  exactly at $R = R_q$  \\cite{Schmid,Bulg}.   \n \nA similar DQPT must exist in the quantum rotator. The analog of the Coulomb energy $e^2\/C$  is the energy \n$\\hbar^2\/ J$ necessary for a transfer of the moment quantum $\\hbar$ to the rotator. The time $\\tau=J\/f$ is the decay time for the moment in the rotator with friction. Thus, the critical friction coefficient is of the order of $f \\sim \\hbar$. \\Eq{Rf} yields the relation between the friction coefficient $f$ and the ohmic resistance $R_r$ for the current produced by the particle of charge $q$ rotating in the rotator. According to this relation, the condition  $f \\sim \\hbar$ for the DQPT in the rotator is identical to the condition $R_r\\sim R_q $. Now $R_q \\sim \\hbar \/q^2$ is the quantum resistance for the charge $q$.\n\n\nHowever, there is a difference in the role of the resistance $R$ in the JJ and of the resistance $R_r$ in the quantum rotator. In the quantum rotator the monotonic phase rotation is possible at small $R_r<R_q$, while in the JJ the phase rotates at large $R>R_q$. In the regime of the phase rotation the JJ is an insulator, while the quantum rotator is a conductor. In the regime of the localized phase the JJ is a superconductor, while the quantum rotator is an insulator. In the both cases the insulating state takes place at  resistances larger than the quantum resistance. \n\n\nOne can estimate the resistance of the quantum rotator using the Drude formula for the conductivity $l\/R_r$, which for the 1D system in the weak-binding limit yields\n\\begin{equation}\n{l\\over R_r}\\sim  {q^2\\tau \\over m_0}={q^2l_0 \\over \\hbar k l},\n    \\ee{}\nwhere $l_0$ is the mean-free path of the particle and $\\tau =l_0\/v=m_0l_0\/\\hbar k$ is the relaxation time at elastic scattering by impurities. Since the wave number $k$ of the particle is on the order of the inverse space period $l$, the phase transition condition \n\\begin{equation}\n{R_q\\over R_r}\\sim {l_0 \\over l^2 k} \\sim kl_0 \\sim 1\n   \\ee{}\nbecomes the Ioffe--Regel condition for the metal--insulator transition \\cite{Mott}.\n\nAlthough the pioneer theoretical investigations \\cite{Schmid,Bulg} predicted the DQPT at the line $R=R_q$ independently from the ratio $E_J\/ E_C$, later it became clear that in the real experiment it is impossible to detect the transition at this line at $E_J\\gg E_C$.  One reason is an inevitable non-zero temperature of the experiment  \\cite{Zaik}. But even at strictly zero temperature the insulator state is not observable at $E_J\\gg E_C$ because  either  the observation time is short compared with the average time interval between tunneling events, which are phase slips destroying superconductivity \\cite{Schon}, or the accuracy of the voltage measurement is not sufficient for detection of the phase rotation since the voltage error bar exceeds the electric breakdown voltage $V_0$ \\cite{SI,Penttila2001}. \n\nThe results of the experimental and theoretical investigation of the phase diagram by Penttil\\\"a {\\em et al.} \\cite{SI,Penttila2001}  are shown in Fig.~\\ref{PD}. Open and black circles show observations of the superconductorlike and the insulatorlike behavior, respectively. The insulating  (I) state differs from the superconducting (S) state by the presence of the Coulomb blockade bump at dependences of the resistance on the current.\nThe solid line was determined from the condition that the error bar of voltage measurements is equal to the voltage $V_0$ at which the electric breakdown of the insulating state takes place. It is impossible to detect the insulating state above the solid line.  The observation of  the Coulomb blockade bump below the solid line  is  a smoking gun of the DQPT.\n\nIf one increases measurement accuracy (or lowers temperature), the  solid line  moves closer to the Schmid--Bulgadaev line. Thus, the vertical  Schmid--Bulgadaev line is an idealized asymptotic limit, which remains experimentally unattainable in practice for large $E_J\/E_C$.  \n\n\\section{Discussion and conclusions}\n\n\n \\begin{figure}[t]\n\\includegraphics[width=.45\\textwidth]{PhD}\n \\caption{The phase diagram of the JJ \\cite{SI}. The dashed vertical line shows the DQPT of  \\citet{Schmid} and \\citet{Bulg}. Due to  voltage measurement error bar the experimental detection of the DQPT is expected at the solid line  \\cite{SI}.\n Black and open symbols show observations of the superconductorlike ($S$) and the insulatorlike ($I$) $RI$ curves (see insets), respectively. Squares shows  results of experimental observations of unshunted junctions when the resistance $R$ is equal to a very large quasiparticle resistance $R_{qp}$ of the junction itself. \\footnote{ In the caption to Fig.~3 in Ref.~\\onlinecite{SI}  it was stated that \"unshunted samples (squares) are collected at  $R_q\/R=0$\". However, as said in the text of the paper, because of a large shunting quasiparticle resistance $R_{qp}$, $R_q\/R$ was never truly equal to 0 in the experiments.}\n \\label{PD}}\n \\end{figure}\n\nAlthough the analogy of the JJ with the quantum rotator was pointed out in the past \\cite{AzbelP,Rogov,widom2,Loss}, the opinion that the JJ is an analog of  a particle in a periodic potential was more prevalent   \\cite{AverLikh,Likh,GefenJ,Zwerg,Apen,SI,Penttila2001,CommMurani,morel,Schon,golub}. Averin, Likharev, and Zorin \\cite{AverLikh,Likh} argued that   the states of the JJ with $\\varphi$ and $\\varphi+2\\pi$ are not identical because the states of the environment (electric circuit) for two values of the phase are different. As a result, they concluded that all states in  the Bloch band are possible and used the concept of wave packets at the derivation  of Bloch oscillations.  \\citet{Zwerg} and  \\citet{morel} explained the distinguishability of the states  with $\\varphi$ and $\\varphi+2\\pi$ (decompactification of the phase) by the effect of dissipation. \\citet{Apen} argued that the distinguishability assumption is justified by taking into account fluctuations of the external classical charge, which were not taken into account in the previous calculations of the $VI$ curve in Refs.~\\cite{widom2,AverLikh,Likh} and in the present paper. \n\n The assumption that the states of the JJ with $\\varphi$ and $\\varphi+2\\pi$ can be different was disputed by \\citet{Loss}. We also believe that the indistinguishability of the states with $\\varphi$ and $\\varphi+2\\pi$ is a just requirement for the quantum rotator and the JJ. Although the connection between the JJ and the environment is of utmost importance and must not be ignored, we consider  the wave function of the junction alone, but not the wave function of the junction+environment.  There is an important difference between two questions: (i) whether the states of the JJ with $\\varphi$ and $\\varphi+2\\pi$ can be different or not, and (ii) whether the states of the JJ and the environment are the same before and after a phase slip, which is  a jump $2\\pi$ of the phase difference across the JJ. \n \nAverin, Likharev, and Zorin \\cite{AverLikh,Likh}  looked for an answer to the second question. Let us illustrate this for a simple SQUID circuit, which  is a superconducting loop interrupted by a JJ. There is a phase $\\varphi$ across the JJ and a phase difference $\\Phi$ along the rest part of the loop. They must satisfy the boundary condition $\\varphi+\\Phi =2\\pi N$, where $N$ is an integer. Averin, Likharev, and Zorin compared the state with $\\varphi$, $\\Phi$,  and $N$,  with  the state with $\\varphi+2\\pi $, $\\Phi-2\\pi$,  and $N$. These are states before and after a phase slip, which are definitely different. While before the phase slip the state was stationary, after the slip the state is not stationary: Adding $2\\pi$ to the Josephson phase does not change the current across the JJ  but  does change the current in the rest part of the circuit and the magnetic flux because the phase difference $\\Phi$ is not the same before and after the slip.\n\nThe states of the JJ with the phases $\\varphi$ and $\\varphi+2\\pi $ must be compared at fixed state of the environment. One should compare  the state with $\\varphi$,  $\\Phi$, and $N$ with the state with $\\varphi+2\\pi $, $\\Phi$, and $N+1$. The environment (the rest part of the circuit) remains in the same state because the phase $\\Phi$  remains the same, and the current and the magnetic flux do not differ.\n \n Through the whole paper it was assumed that the external moment in the quantum rotator (charge in the JJ) is a well defined classical variable slowly varying in time.  As mentioned above,\n fluctuations of the external charge were considered as a reason for the phase decompactification in the JJ. Indeed, for any {\\em given} external charge only {\\em one} state in the Bloch band  is possible, but this state is different for different  external charges. Thus, for a broad ensemble of external charges any state in the Bloch band  is possible as in the case of a particle in the infinite 1D state. However, this does not eliminate the difference between the Hilbert space  of all possible Bloch states in the case of a particle in the infinite 1D state and its much smaller discrete subspace of the rotator states with quantized canonical moments (charges). If one has the ensemble of $n$ external moments, the number of states of the  system ``quantum rotator+environment'' is equal to $n$, while  the number of states for ``particle in an infinite 1D space+environment'' is $n\\times n_B$, where $n_B$ is the number of all Bloch states in a Bloch band.\n  \n This difference is not important for the conventional theory on the insulating state and the DQPT. This is true because the theory deals only with a slow adiabatic tuning of only {\\em one} state in the lowest Bloch band. The question whether this state is a single state or other Bloch states  are possible, is irrelevant. This is another example when  according to \\citet{Legg} ``it is largely unnecessary  to address the vexed question of whether or not states differing in $\\varphi$ value by $2\\pi$ should be identified.''\n \n This does not mean that difference between dynamics of a particle in a 1D ring and a particle in the infinite 1D space is not important in general, at any experimental conditions. \n \\citet{Loss} made calculations of the resonant tunneling between quantum levels in the neighboring potential wells of the washboard potential in the JJ.  In the Bloch theory this is the Zener interband tunneling. The results depended on the choice of the initial state. The latter  was either periodic in the extended phase as must be for a particle in a 1D ring, or was confined in the interval $2\\pi$ that is possible only for a particle in the infinite 1D space. This calculation has not yet resolved the dilemma ``compact vs. extended phase''. \\citet{Loss}  solved the Schr\\\"odinger equation \\eq{ShQ} at the constant current $I= \\dot Q_0$, while the proponents of the extended phase assumption for the JJ connected the decompactification with fluctuations of the external charge $Q_0$, which were not taken into account in their calculation. Thus,  the answer to the question whether and how the difference between motion of a particle in a 1D ring and motion of a particle in the infinite 1D space with a periodic potential can affect observable phenomena  is still lacking.   \n\n\nArguing that the conventional theory failed and the insulating state is impossible in the JJ, \\citet{Sacl}  referred to the existence of supercurrent in the Cooper pair box, which is identical to a JJ in the limit $R\\to \\infty$. Indeed, in Sec.~\\ref{rcf}  we demonstrated that without dissipation the phase (angle) of the quantum rotator is localized and  performs Bloch oscillations around the localization point. Thus, a supercurrent flows across the JJ. However, from the same subsection it is clear that {\\em any} dissipation whatever large $R$ be,  delocalizes the phase and the JJ becomes an insulator. One can see in Fig.~\\ref {f1.1} that at $R \\to \\infty $ the voltage vanishes at currents $I\\gg e\/RC^*$. Although the insulating state is possible at any $R$, the electric breakdown  of the insulator occurs at the current  $I\\sim e\/RC^*$, which vanishes at $R \\to \\infty $. This makes observation of the DQPT impossible in this limit, but it is not an argument against its existence. \n\n\nSummarizing, our answers to three questions formulated in the introduction are following:\n\\begin{enumerate}\n\\item\nThe states with the phases $\\varphi$ and $\\varphi+2\\pi$ are indistinguishable in the JJ for the fixed state of the environment. In this aspect the JJ is an analog of the quantum rotator, but not of a particle in a periodic potential.\n\\item\nThe assumption that in the quantum rotator and in the JJ the states  with the phases $\\varphi$ and $\\varphi+2\\pi$ are indistinguishable, does not mean that the wave functions must be also periodic in all gauges. The wave functions can be  periodic only in the periodic gauge where the Hamiltonian is periodic in $\\varphi$. \n\\item\nThe conventional  theory of the DQPT is valid independently from whether the states with the phases $\\varphi$ and $\\varphi+2\\pi$ are  distinguishable, or not.\n\\end{enumerate}\n\nThe present analysis and its conclusions referred only to the case of a single particle, which excludes  interaction between particles. The case of many interacting particles in a quantum rotator requires another approach to the dilemma ``compact vs. extended  phase'' \\cite{averin}.\n\n\nThe close analogy between the regime of phase rotation in the quantum  rotator and the phase rotation in the insulating state of  the JJ allows to  expect the DQPT also in the quantum rotator. This can be checked by experimental investigations of persistent currents in 1D normal rings put in a constant electric field.   \n\n\n\n\n\n\n\\begin{acknowledgments}\nI thank Dmitri Averin, Pertti  Hakonen, and Andrei Zaikin for discussions and useful comments.\n\\end{acknowledgments}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{XP algorithm parameterized by neighborhood diversity}\\label{sec:nd}\nIn this section, we deal with neighborhood diversity ${\\nu}$, which is a more general parameter than vertex cover number. We first give an $O^*((n\/{\\nu})^{{\\nu}})$-time algorithm for Arc Kayles. This is an XP algorithm parameterized by neighborhood diversity. On the other hand, we show that there is a graph having at least $O^*((n\/{\\nu})^{{\\rm \\Omega}({\\nu})})$ non-isomorphic induced subgraphs, which implies the analysis of the proposed algorithm is\nasymptotically tight. \n\n\nBy Proposition \\ref{prop:isomorphic}, if we list up all non-isomorphic induced subgraphs, the winner of Arc Kayles can be determined by using recursive formulas (\\ref{eq:recursion1}) and (\\ref{eq:recursion2}).\nLet $\\mathcal{M}=\\{M_1,M_2,\\ldots,M_{{\\nu}}\\}$ be a partition such that $\\bigcup_i M_i=V$ and vertices of $M_i$ are twins each other.  We call each $M_i$ a \\emph{module}.\nWe can see that non-isomorphic induced subgraphs of $G$ are identified by how many vertices are selected from which module.\n\n\\begin{lemma}\\label{lem:positions:nd}\nThe number of non-isomorphic induced subgraphs of a graph of neighborhood diversity ${\\nu}$ is at most $(n\/{\\nu}+1)^{{\\nu}}$.\n\\end{lemma}\n\\begin{proof}\nBy the definition of neighborhood diversity, vertices in a module are twins each other. Therefore, the number of non-isomorphic induced subgraphs of $G$ is at most \n    $\\prod_{i=1}^{{\\nu}}(\\card{M_i}+1) \\le (\\sum_{i=1}^{{\\nu}}(\\card{M_i}+1)\/{\\nu})^{{\\nu}}\n    \\le (n\/{\\nu}+1)^{{\\nu}}$.\n   \n\\end{proof}\n\nWithout loss of generality, we select an edge whose endpoints are the minimum indices of vertices in the corresponding module. \nBy Proposition \\ref{prop:isomorphic}, the algorithm in Section \\ref{sec:Exalgo} can be modified to run in time $O^*((n\/{\\nu}+1)^{{\\nu}})$.\n\n\n \n\\begin{theorem}\\label{thm:algo:nd}\nThere is an $O^*((n\/{\\nu}+1)^{{\\nu}})$-time algorithm for Arc Kayles.\n\\end{theorem}\n \nThe idea can be extended to Colored Arc Kayles and BW-Arc Kayles. \nIn $G=(V,E_{\\CG}\\cup E_{\\CB}\\cup E_{\\CW})$, two vertices $u,v\\in V$ are called \\emph{colored twins} if $c(\\{u,w\\})=c(\\{v,w\\})$ holds $\\forall w \\in V\\setminus \\{u,v\\}$. \nWe then define the notion of colored neighborhood diversity. \n\\begin{definition}\\label{def:cnd}\nThe \\emph{colored neighborhood diversity} of $G=(V,E)$ is defined as minimum ${\\nu'}$ such that $V$ can be partitioned into ${\\nu'}$ vertex sets of colored twins.\n\\end{definition} \nIn Colored Arc Kayles or BW-Arc Kayles, we can utilize a partition of $V$ into modules each of which consists of colored twins. \nIf we are given a partition of the vertices into colored modules, we can decide the winner of Colored Arc Kayles or BW-Arc Kayles like \nTheorem \\ref{thm:algo:nd}. Different from ordinary neighborhood diversity,  it might be hard to compute colored neighborhood diversity in polynomial time. \n\\begin{theorem}\\label{thm:algo:cnd}\nGiven a graph $G=(V,E_{\\CG}\\cup E_{\\CB}\\cup E_{\\CW})$ with a partition of $V$ into ${\\nu'}$ modules of colored twins, we can compute the winner of Colored Arc Kayles on $G$ in time $O^*((n\/{\\nu'}+1)^{{\\nu'}})$.   \n\\end{theorem}\n\n\n\n    \n\n  \n     \n \n \nIn the rest of this section, we give a bad instance for the proposed algorithm as shown in Figure \\ref{fig:lower:nd}.\nThe result implies that the analysis of Theorem \\ref{thm:algo:nd} is\nasymptotically tight. \n\n\\begin{theorem}\\label{thm:lower:nd}\nThere is a graph having at least $(n\/{\\nu}+1-o(1))^{{\\nu}(1 - o(1))}$ non-isomorphic positions of Arc Kayles. \n\\end{theorem}\n\n\\begin{proof}\nWe construct such a graph $G$. Assume that $k$ is a number forming power of two minus one, that is, $k=2^{k'}-1$. \nFirst, we prepare $k$ cliques of $s$ vertices, $C_1,\\ldots,C_k$, and vertex set $X=\\{x_1,x_2,\\ldots,x_{\\log_2 (k+1)}\\}$. The subscript $i$ of $x_i$ represents $i$-th bit used below. \nFor each $x_i$, we attach $i-1$ pendant vertices, which is used to distinguish from another $x_j$. For $j$, $\\mathrm{bin}(j)$ and $\\mathrm{bin}(j,i)$ denote the $j$'s binary representation and its $i$-th bit, respectively. For example, $\\mathrm{bin}(6)=110$, \n$\\mathrm{bin}(6,1)=0, \\mathrm{bin}(6,2)=1, \\mathrm{bin}(6,3)=1$, and $\\mathrm{bin}(6,i)=0$ for $i\\ge 4$. \nWe connect the vertices in $C_j$ and vertices in $X$ according to the binary representation of $j$; vertices in $C_j$ are connected with $x_i$ if and only if $\\mathrm{bin}(j,i)=1$. Finally, we connect all the vertices in $\\bigcup_i C_i$, which form a large clique with size $sk$. \nFigure \\ref{fig:lower:nd} shows the constructed graph $G$.\n\nThe number of vertices in $G$ is $n=sk + \\log_2 (k+1)(\\log_2 (k+1)+1)\/2$, that is, $s=(n-\\log_2 (k+1)(\\log_2 (k+1)+1)\/2)\/k$, \nand the neighborhood diversity of $G$ is ${\\nu}=k+2\\log_2 (k+1)$, \nbecause vertices in each clique are twins, and also pendant vertices connected to each $x_i$ are twins. \n\n\n\nWe estimate the number of non-isomorphic induced subgraphs of $G$. \nWe restrict vertices to delete only from $\\bigcup_i C_i$, \nthat is, edges to select are inside of $\\bigcup_i C_i$. \nSince the number of pendant vertices for each $x_i$ is different, \n$x_i$'s substantially have IDs, and thus vertices from two distinct cliques are distinguishable. Hence, the number of non-isomorphic induced subgraphs obtained by removing edges inside $\\bigcup_i C_i$ are decided by the numbers of remaining vertices in $C_i$'s, each of which varies from $0$ to $s$. \nTherefore, the number of non-isomorphic induced subgraphs is at least\n\\begin{align*}\n(s+1)^{k}\/2 &= \\frac{1}{2}\\left(\\frac{n-\\log_2 (k+1)(\\log_2 (k+1)+1)\/2}{k}+1\\right)^{k}\\\\\n& = \\frac{1}{2}\\left(\\frac{n-o(k)}{k}+1\\right)^{k} \\ge \n\\frac{1}{2}\\left(\\frac{n}{k}+1-o(1)\\right)^{k},\n\\end{align*}\nwhere the division of $2$ comes from the fact that the number of deleted vertices must be even. \nSince ${\\nu}=k+2\\log_2 (k+1)$, $k={\\nu} - 2\\log_2 (k+1)\\ge {\\nu} - 2\\log_2  {\\nu}$ holds. We thus have lower bound $\\left(n\/{\\nu}+1-o(1)\\right)^{{\\nu}(1 - o(1))}$. \n\\end{proof}\n\n\\begin{figure}[tbp]\n \\centering\n \\includegraphics[width=0.65\\linewidth]{nd_lowler.pdf}\n \\caption{The constructed graph $G$ with neighborhood diversity ${\\nu}=k+2\\log_2 (k+1)$.}\n \\label{fig:lower:nd}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section\nBasic Algorithm }\\label{sec:Exalgo}\nIn this section, we show that the winner of \\emph{Colored} Arc Kayles on $G$ can be determined in time $O^*(2^n)$. We first observe that  the following lemma holds by the definition of the game.\n\\begin{lemma}\\label{lem:winner:Colored_Arc_Kayles}\nSuppose that Colored Arc Kayles is played on $G=(V,E_{\\CG}\\cup E_{\\CW}\\cup E_{\\CB})$. \nThen, player $\\mathrm{B}$ (resp., $\\mathrm{W}$) wins on $G$ with player $\\mathrm{B}$'s (resp., $\\mathrm{W}$'s) turn if and only if there is an edge $\\{u,v\\}\\in E_{\\CG}\\cup E_{\\CB}$ (resp., $\\{u,v\\}\\in E_{\\CG}\\cup E_{\\CW}$) such that player $\\mathrm{W}$ (resp., $\\mathrm{B}$) loses on $G-u-v$ with player $\\mathrm{W}$'s (resp., $\\mathrm{B}$'s) turn. \n\\end{lemma}\nThis lemma is interpreted by the following two recursive formulas:\n\\begin{align}\\label{eq:recursion1}\n    f_{\\mathrm{B}}(G) & = \\bigvee_{\\{u,v\\}\\in{E_{\\CG}\\cup E_{\\CB}}}\\lnot \\left(f_{\\mathrm{W}}(G-u-v)\\right),\\\\ \\label{eq:recursion2} \n    f_{\\mathrm{W}}(G) & = \\bigvee_{\\{u,v\\}\\in{E_{\\CG}\\cup E_{\\CW}}}\\lnot \\left(f_{\\mathrm{B}}(G-u-v)\\right).      \n\\end{align}\nBy these formulas, we can determine the winner of $G$ with either first or second player's turn by computing $f_{\\mathrm{B}}(G)$ and $f_{\\mathrm{W}}(G)$ for all induced subgraphs of $G$. Since the number of all induced subgraphs of $G$ is $2^n$,  it can be done in time $O^*(2^n)$ by a standard dynamic programming algorithm. \n\n\\begin{comment}\nOur algorithm is based on dynamic programming with respect to induced subgraphs of an input graph $G$. \nAs a key observation, any position of Colored Arc Kayles corresponds to an induced subgraph of $G$.\n\nFurther, we have the following lemma.\n\nBased on Lemma \\ref{lem:winner:Colored_Arc_Kayles}, we present Algorithm \\ref{alg:Colored_Arc_Kayles}. The input of Algorithm \\ref{alg:Colored_Arc_Kayles} is a graph $G=(V,E_{\\CG}\\cup E_{\\CB}\\cup E_{\\CW})$ and three edge sets $E_{\\CG}$,$E_{\\CB}$, and $E_{\\CW}$. Algorithm \\ref{alg:Colored_Arc_Kayles} returns 1 if the first player, which can choose edges in $E_{\\CG}\\cup E_{\\CB}$, wins. Otherwise, it returns 0. In the algorithm, if $E_{\\CG}\\cup E_{\\CB}$ is empty, the first player $p_1$ cannot choose an edge, and hence he\/she loses. Otherwise, he\/she picks an edge from $E_{\\CG}\\cup E_{\\CB}$. By Lemma \\ref{lem:winner:Colored_Arc_Kayles}, the winner is determined on $G-u-v$. \nHere, the second player $p_2$ loses on $G-u-v$ if and only if there is no edge $\\{u',v'\\}\\in E_{\\CG}\\cup E_{\\CW}\\setminus\\Gamma(\\{u,v\\})$ such that $p_1$ loses on $G-u-v-u'-v'$. That is, the winner on $G$ can be found by considering a game such that the first player is $p_2$, the second player is $p_1$, and the input graph is $G'-u-v$. Thus, in Line 3 of Algorithm \\ref{alg:Colored_Arc_Kayles}, it returns $\\bigvee_{\\{u,v\\}\\in{E_{\\CG}\\cup E_{\\CB}}}\\lnot \\left({\\bf Colored\\_Arc\\_Kayles}(G-u-v)\\right)$. \n\nSince the number of all induced subgraphs is at most $2^n$, the running time of Algorithm \\ref{alg:Colored_Arc_Kayles} is $O^*(2^n)$.\n\n\\begin{algorithm*}[t]\n     \\caption{${\\bf Colored\\_Arc\\_Kayles}(G,E_{\\CG},E_{\\CW},E_{\\CB})$, {\\bf Input:} $G=(V,E_{\\CG}\\cup E_{\\CW}\\cup{E_{\\CB}})$}\n     \\begin{algorithmic}[1]\\label{alg:Colored_Arc_Kayles}\n\n     \\IF {$E_{\\CG}\\cup E_{\\CB}=\\emptyset$}\n        \\RETURN $0$\n\n     \\ELSE\n         \\RETURN \\\\\n         $\\bigvee_{\\{u,v\\}\\in{E_{\\CG}\\cup E_{\\CB}}}\\lnot \\left({\\bf Colored\\_Arc\\_Kayles}(G-u-v)\\right)$\n     \\ENDIF\n \\end{algorithmic}\n \\end{algorithm*}\n\\end{comment}\n\n\\begin{theorem}\\label{thm:Colored_Arc_Kayles:Exp}\nThe winner of Colored Arc Kayles can be determined in time $O^*(2^n)$.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\subsection{Background and Motivation} \nCram, Domineering, and Arc Kayles are well-studied two-player mathematical games and \ninterpreted as combinatorial games on graphs. \nDomineering (also called Stop-Gate) was   \nintroduced by G\\\"oran Andersson around 1973 under the name of Crosscram~\\cite{conway2000numbers,gardner1974mathematical}.\nDomineering is usually played on a checkerboard. The two players are denoted by\nVertical and Horizontal. Vertical (resp., Horizontal) player is\nonly allowed to place its dominoes vertically (resp., horizontally) on the board.\nNote that placed dominoes are not allowed to overlap. \nIf no place is left to place a domino, the player in the turn loses the game. Domineering is a partisan game, where players use different pieces. The impartial version of the game is Cram, where two players can place dominoes both vertically and horizontally. \n\nAn analogous game played on an undirected graph $G$ is Arc Kayles.\nIn Arc Kayles, the action of a player in a turn is to select an edge of $G$, and then the selected edge and its neighboring edges are removed from $G$. If no edge remains in the resulting graph, the player in the turn loses the game.  \nFigure \\ref{ex:arcKayles} is a play example of Arc Kayles. In this example, the first player selects edge $e_1$, and then the second player selects edge $e_2$. By the first player selecting edge $e_3$, no edge is left; the second player loses.  \nNote that the edges selected throughout a play form a maximal matching on the graph.\n\nSimilarly, we can define BW-Arc Kayles, which is played on an undirected graph with black and white edges. The rule is the same as the ordinary Arc Kayles except that the black (resp., white) player can select only black (resp., white) edges. \nNote that Cram and Domineering are respectively interpreted as Arc Kayles and BW-Arc Kayles on a two-dimensional grid graph, which is the graph Cartesian product of two path graphs.  \n\nTo focus on the common nature of such games with matching structures, we newly define {Colored Arc Kayles}. {Colored Arc Kayles} is played on a graph whose edges are colored in black, white, or gray, and black (resp., white) edges can be selected only by the black (resp., white) player, though grey edges can be selected by both black and white players. \n{BW-Arc Kayles} and ordinary {Arc Kayles} are special cases of {Colored Arc Kayles}. In this paper, we investigate {Colored Arc Kayles} from the algorithmic point of view.\n\n\\begin{figure}[tbp]\n \\centering\n \\includegraphics[width=0.8\\linewidth]{arc_kayles.pdf}\n \\caption{A play example of Arc Kayles}\n \\label{ex:arcKayles}\n\\end{figure}\n\n\\subsection{Related work}\n\\subsubsection{Cram and Domineering}\nCram and Domineering are well studied in the field of combinatorial game theory. In \\cite{gardner1974mathematical}, Gardner gives winning strategies for some simple cases. For Cram on $a\\times b$ board, the second player can always win if both $a$ and $b$ are even, and the first player can always win if one of $a$ and $b$ is even and the other is odd. This can be easily shown by the so-called Tweedledum and Tweedledee strategy. \nFor specific sizes of boards, computational studies have been conducted~\\cite{uiterwijk2019solving}. \nIn \\cite{uiterwijk2018construction}, Cram's endgame databases for all board sizes with at most 30 squares are constructed. \nAs far as the authors know, the complexity to determine the winner for Cram on general boards still remains open. \n\n\nFinding the winning strategies of Domineering for specific sizes of boards by using computer programs is well studied. \nFor example, the cases of $8\\times 8$ and $10\\times 10$ are solved in 2000~\\cite{BREUKER2000195} and 2002~\\cite{bullock2002domineering}, respectively. The first player wins in both cases. Currently, the status of boards up to $11\\times 11$ is known~\\cite{uiterwijk201611}. \nIn \\cite{uiterwijk2015new}, endgame databases for all single-component positions up to 15 squares for Domineering are constructed.\nThe complexity of Domineering on general boards also remains open. Lachmann, Moore, and Rapaport show that the winner and a winning strategy Domineering on $m\\times n$ board can be computed in polynomial time for $m\\in \\{1, 2, 3, 4, 5, 7, 9, 11\\}$ and all $n$~\\cite{lachmann2000wins}.\n\n\n\\subsubsection{Kayles, Node Kayles, and Arc Kayles}\nKayles is a simple impartial game, introduced by Henry Dudeney in 1908~\\cite{dudeney2002canterbury}.  \nThe name ``Kayles'' derives from French word ``quilles'', meaning ``bowling''. The rule of Kayles is as follows.  \nGiven bowling pins equally spaced in a line, players take turns to knock out either one pin or two adjacent pins, until all the pins are gone.\nAs graph generalizations, Node Kayles and Arc Kayles are introduced by Schaefer~\\cite{SCHAEFER1978185}. \nNode Kayles is the vertex version of Arc Kayles. Namely, \nthe action of a player is to select a vertex instead of an edge, and then the selected vertex and its neighboring vertices are removed. \nNote that both generalizations can describe the original Kayles; Kayles is represented as Node Kayles on sequentially linked triangles or as Arc Kayles on a caterpillar graph.\n\nNode Kayles is known to be PSPACE-complete \\cite{SCHAEFER1978185}, whereas the winner determination is solvable in polynomial time on graphs of bounded asteroidal numbers such as cocomparability graphs and cographs by using Sprague-Grundy theory~\\cite{BODLAENDER2002}. \nFor general graphs, Bodlaender et al. propose an $O(1.6031^n)$-time algorithm~\\cite{BODLAENDER2015165}. Furthermore, they show that the winner of Node Kayles can be determined in time $O(1.4423^{n})$ on trees. In \\cite{Kobayashi2018}, Kobayashi sophisticates the analysis of the algorithm in \\cite{BODLAENDER2015165} from the perspective of the parameterized complexity and shows that it can be solved in time $O^*(1.6031^{{\\mathtt{\\mu}}})$, where ${\\mathtt{\\mu}}$ is the modular width of an input graph\\footnote[1]{The $O^*(\\cdot)$ notation suppresses polynomial factors in the input size.}. He also gives an $O^*(3^{{\\tau}})$-time algorithm, where ${\\tau}$ is the vertex cover number, \nand a linear kernel when parameterized by neighborhood diversity.\n\nDifferent from Node Kayles, the complexity of Arc Kayles has remained open for more than 30 years. Even for subclasses of trees, not much is known. For example, Huggans and Stevens study Arc-Kayles on subdivided stars with three paths~\\cite{huggan2016polynomial}. \nTo our best knowledge,\nno exponential-time algorithm for Arc Kayles is presented except for an $O^*(4^{{\\tau}^2})$-time algorithm proposed in \\cite{LM2014}.\n\n\\subsection{Our contribution}\n\nIn this paper, we address winner determination algorithms for Colored Arc Kayles. \nWe first propose an $O^*(2^n)$-time algorithm for Colored Arc Kayles.\nNote that this is generally faster than applying the Node Kayles algorithm  to the line graph of an instance of Arc Kayles;  \nit takes time $O(1.6031^m)$, where $m$ is the number of the original edges.\nWe then focus on algorithms based on graph parameters.\nWe present an $O^*(1.4143^{{\\tau}^2+3.17{\\tau}})$-time algorithm for Colored Arc Kayles, where ${\\tau}$ is the vertex cover number. \nThe algorithm runs in time $O^*(1.3161^{{\\tau}^2+4{\\tau}})$ and $O^*(1.1893^{{\\tau}^2+6.34{\\tau}})$ for BW-Arc Kayles, and Arc Kayles, respectively.\nThis is faster than the previously known time complexity $O^*(4^{{\\tau}^2})$ in  \\cite{LM2014}. \n\nOn the other hand, we give a bad instance for the proposed algorithm, which implies the running time analysis is asymptotically tight.\nFurthermore, we show that the winner of Arc Kayles can be determined in time $O^*((n\/{\\nu}+1)^{{\\nu}})$,  where ${\\nu}$ is the neighborhood diversity of an input graph. This analysis is also asymptotically tight, because there is an instance having $(n\/{\\nu}-o(1))^{{\\nu}(1-o(1))}$.\nWe finally show that the winner determination of Arc Kayles on trees can be solved in $O^*(2^{n\/2})=O(1.4143^n)$ time, which improves $O^*(3^{n\/3})(=O(1.4423^n))$ by a direct adjustment of the analysis of Bodlaender et al.'s $O^*(3^{n\/3})$-time algorithm for Node Kayles. \n\n\n\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\\label{sec:preliminaries}\n\\subsection{Notations and terminology}\nLet $G=(V,E)$ be an undirected graph. We denote $n=\\card{V}$ and $m=\\card{E}$, respectively. \nFor an edge $e=\\{u,v\\}\\in E$, we define $\\Gamma(e)=\\{e'\\mid e\\cap e'\\neq \\emptyset\\}$.\nFor a graph $G=(V,E)$ and a vertex subset $V'\\subseteq V$, we denote by $G[V']$ the subgraph induced by $V'$. For simplicity, we denote $G-v$ instead of $G[V\\setminus \\{v\\}]$. For an edge subset $E'$, we also denote by $G-E'$ the subgraph obtained from $G$ by removing all edges in $E'$ from $G$. \nA vertex set $S$ is called a \\emph{vertex cover} if  $e\\cap S\\neq \\emptyset$ for every edge $e\\in E$. We denote by ${\\tau}$ the size of a minimum vertex cover of $G$.\nTwo vertices $u,v\\in V$ are called \\emph{twins} if $N(u)\\setminus \\{v\\}=N(v)\\setminus \\{u\\}$.\n\n\\begin{definition}\\label{def:nd}\nThe \\emph{neighborhood diversity} ${\\nu}(G)$ of $G=(V,E)$ is defined as the minimum number $w$ such that $V$ can be partitioned into $w$ vertex sets of twins. \n\\end{definition}\n\nIn the following, we simply write ${\\nu}$ instead of ${\\nu}(G)$ if no confusion arises. We can compute the neighborhood diversity of $G$ and the corresponding partition in polynomial time \\cite{Lampis2012}. For any graph $G$,  ${\\nu} \\le 2^{{\\tau}}+{\\tau}$ holds.\n\n\\subsection{Colored Arc Kayles}\nColored Arc Kayles is played on a graph $G=(V,E_{\\CG}\\cup E_{\\CB}\\cup E_{\\CW})$, where $E_{\\CG},E_{\\CB}, E_{\\CW}$ are mutually disjoint. The subscripts $\\mathrm{G}$, $\\mathrm{B}$, and $\\mathrm{W}$ of $E_{\\CG}, E_{\\CB}, E_{\\CW}$ respectively, stand for gray, black, and white. \nFor every edge $e\\in E_{\\CG} \\cup E_{\\CB} \\cup E_{\\CW}$, let $c(e)$ be the color of $e$, that is, $c(e)=\\mathrm{G}$ if $e\\in E_{\\CG}$, $\\mathrm{B}$ if $e\\in E_{\\CB}$, and $\\mathrm{W}$ if $e\\in E_{\\CW}$.\nIf $\\{u,v\\}\\not\\in E_{\\CG} \\cup E_{\\CB} \\cup E_{\\CW}$, we set $c(\\{u,v\\})=\\emptyset$ \nfor convenience. \nAs explained below, the first (black or $\\mathrm{B}$) player can choose only gray or black edges, and the second (white or $\\mathrm{W}$) player can choose only gray or white edges. \n\nTwo players alternatively choose an edge of $G$. Player B can choose an edge in $E_{\\CG}\\cup E_{\\CB}$ and player W can choose an edge in $E_{\\CG}\\cup E_{\\CW}$.\nThat is, there are three types of edges; $E_{\\CB}$ is the set of edges that only the first player can choose, $E_{\\CW}$ is the set of edges that only the second player can choose, and $E_{\\CG}$ is the set of edges that both the first and second players can choose. Once an edge $e$ is selected, the edge and its neighboring edges (i.e., $\\Gamma(e)$) are removed from the graph, and the next player chooses an edge of $G- \\Gamma(e)$.  \nThe player that can take no edge loses the game. \nSince (Colored) Arc Kayles is a two-person zero-sum perfect information game and ties are impossible, one of the players always has a winning strategy. \nWe call the player having a winning strategy the \\emph{definite winner}, or simply \\emph{winner}. \n\nThe problem that we consider in this paper is defined as follows: \n\\begin{description}\n\\item {\\bf Input}: $G=(V,E_{\\CG}\\cup E_{\\CB}\\cup E_{\\CW})$, active player in $\\{\\mathrm{B},\\mathrm{W}\\}$. \n\\item {\\bf Question}: Suppose that players $\\mathrm{B}$ and $\\mathrm{W}$ play Colored Arc Kayles on $G$ from the active player's turn. Which player is the winner?\n\\end{description}\nRemark that if $E_{\\CB}=E_{\\CW}=\\emptyset$, Colored Arc Kayles is equivalent to Arc Kayles and if $E_{\\CG}=\\emptyset$, it is equivalent to BW-Arc Kayles.\n\nTo simply represent the definite winner of Colored Arc Kayles, we introduce two Boolean functions $f_{\\mathrm{B}}$ and $f_{\\mathrm{W}}$. The $f_{\\mathrm{B}}(G)$ is defined such that $f_{\\mathrm{B}}(G)=1$ if and only if the winner of Colored Arc Kayles on $G$ from player B's turn is player B. Similarly, $f_{\\mathrm{W}}(G)$ is the function such that $f_{\\mathrm{W}}(G)=1$ if and only if the winner of Colored Arc Kayles on $G$ from player W's turn is the player W. If two graphs $G$ and $G'$ satisfy that \n$f_{\\mathrm{B}}(G)=f_{\\mathrm{B}}(G')$ and $f_{\\mathrm{W}}(G)=f_{\\mathrm{W}}(G')$, we say that \\emph{$G$ and $G'$ have the same game value on Colored Arc Kayles}.    \n\n\n\n\\section{Arc Kayles for Trees}\\label{sec:trees}\nIn \\cite{BODLAENDER2015165}, Bodlaender et al. show that the winner of Node Kayles on trees can be determined in time $O^*(3^{n\/3})=O(1.4423^n)$. \n It is easy to show by a similar argument that the winner of Arc Kayles can also be determined in time $O(1.4423^n)$. \n It is also mentioned that the analysis is sharp apart from a polynomial factor because there is a tree for which the algorithm takes $\\Omega(3^{n\/3})$ time. The example is also available for Arc Kayles; namely, as long as we use the same algorithm, the running time cannot be improved.  \n \n \\medskip\n \nIn this section, we present that the winners of Arc Kayles on trees can be determined in time $O^*(2^{n\/2})=O(1.4143^n)$, which is attained by considering a tree (so-called) unordered. Since a similar analysis can be applied to Node Kayles on trees, the winner of Node Kayles on trees can be determined in time $O^*(2^{n\/2})$.\n\n\\medskip \n\n \n\nLet us consider a tree $T=(V,E)$. \nBy Sprague\u2013Grundy theory, if all connected subtrees of $T$ are enumerated, one can determine the winner of Arc Kayles. \nFurthermore, by Proposition \\ref{prop:isomorphic}, once a connected subtree $T'$ is listed, we can ignore subtrees isomorphic to $T'$. Here we adopt \nisomorphism of rooted trees. \n\\begin{definition}\\label{def:isomorphic:tree}\nLet $T^{(1)}=(V^{(1)},E^{(1)},r^{(1)})$ and $T^{(2)}=(V^{(2)},E^{(2)},r^{(2)})$ be trees rooted at $r^{(1)}$ and $r^{(2)}$, respectively. Then,  \n$T^{(1)}$ and $T^{(2)}$ are called \\emph{isomorphic with respect to root} if for any pair of $u,v\\in V^{(1)}$ there is a bijection $f: V^{(1)}\\to V^{(2)}$ such that $\\{u,v\\}\\in E^{(1)}$ if and only if $\\{f(u),f(v)\\}\\in E^{(2)}$ and $f(r^{(1)})=f(r^{(2)})$. \n\\end{definition}\nFor a tree $T$ rooted at $r$, two subtrees $T'$ and $T''$ are simply said \\emph{non-isomorphic} \nif $T'$ with root $r$ and $T''$ with root $r$ are not isomorphic with respect to root. \nNow, we estimate the number of non-isomorphic connected subgraphs of $T$ based on isomorphism of rooted trees. \nFor $T=(V,E)$ rooted at $r$, a connected subtree $T'$ rooted at $r$ is called an \\emph{AK-rooted subtree} of $T$, if there exists a matching $M  \\subseteq E$ such that $T[V\\setminus \\bigcup M]$ consists of $T'$ and isolated vertices.\nNote that $M$ can be empty, AK-rooted subtree $T'$ must contain root $r$ of $T$, and the graph consisting of only vertex $r$ can be an AK-rooted subtree.\n\n\n\\begin{lemma}\\label{lem:K-set:tree}\nAny tree rooted at $r$ has $O^*(2^{n\/2})(=O(1.4143^n))$ non-isomorphic AK-rooted subtrees rooted at $r$.  \n\\end{lemma}\n\n\\begin{proof}\n Let $R(n)$ be the maximum number of non-isomorphic AK-rooted subtrees of any tree rooted at some $r$ with $n$ vertices. \nWe claim that $R(n)\\le 2^{n\/2}-1$ for all $n\\geq 4$, which proves the lemma. \n\nWe will prove the claim by induction. For $n\\le 4$, the values of $R(n)$'s are as follows: $R(1)=1, R(2)=1, R(3)=2$, and $R(4)=3$. These can be shown by concretely enumerating trees. For example, for $n=2$, a tree $T$ with $2$ vertices is unique, and an AK-rooted subtree of $T$ containing $r$ is also unique, which is $T$ itself. For $n=3$, the candidates of $T$ are shown in Figure \\ref{tree1}. For Type A in Figure \\ref{tree1}, AK-rooted subtrees are the tree itself and isolated $r$, and for Type B, an AK-rooted subtree is only the tree itself; thus we have $R(3)=2$. Similarly, we can show $R(4)=3$ as seen in Figure \\ref{tree2}.\nNote that $R(1)>2^{1\/2}-1$, $R(2)=1\\le 2^{2\/2}-1=1$, $R(3)=2 > 2^{3\/2}-1$, and $R(4)=3\\le 2^{4\/2}-1=3$. This $R(4)$ is used as the base case of induction. \n\n\\begin{figure}[htbp]\n\\begin{minipage}{0.48\\hsize}\n \\centering\n \\includegraphics[width=0.55\\linewidth]{tree3.pdf}\n \\caption{Trees with 3 vertices rooted at $r$}\n \\label{tree1}\n\\end{minipage}\n\\begin{minipage}{0.48\\hsize}\n \\centering\n \\includegraphics[width=1.05\\linewidth]{tree4.pdf}\n \\caption{Trees with 4 vertices rooted at $r$}\n \\label{tree2}\n\\end{minipage}\n\\end{figure}\nAs the induction hypothesis, let us assume that the claim is true \nfor all $n' < n$ except $1$ and $3$, and consider a tree $T$ rooted at $r$ with $n$ vertices. Let $u_1, u_2, \\ldots, u_p$ be the children of root $r$, and $T_i$ be the subtree of $T$ rooted at $u_i$ with $n_i$ vertices for $i=1,2,\\ldots,p$. \nNote that for an AK-rooted subtree $T'$ of $T$, the intersection of $T'$ and $T_i$ for each $i$ is either empty or an AK-rooted subtree of $T_i$ rooted at $u_i$. Based on this observation, we take a combination of the number of AK-rooted subtrees of $T_i$'s, which gives an upper bound on the number of AK-rooted subtrees of $T$. We consider two cases: (1) for any $i$, $n_i \\neq 3$, (2) otherwise. For case (1), the number of AK-rooted subtrees of $T$ is at most \n\\[\n\\prod_{i:n_i>1}(R(n_i)+1)\\cdot \\prod_{i:n_i=1}1 \\le \\prod_{i:n_i>1} 2^{n_i\/2}=2^{\\sum_{i:n_i>1} n_i\/2} \\le 2^{(n-1)\/2} \\le 2^{n\/2}-1.    \n\\]\nThat is, the claim holds in this case. Here, in the left hand of the first inequality, $R(n_i)+1$ represents the choice of AK-rooted subtree of $T_i$ rooted at $u_i$ or empty, \nand ``$1$'' for $i$ with $n_i=1$ represents that $u_i$ needs to be left as is, because otherwise edge $\\{r,u_i\\}$ must be removed, which violates the condition ``rooted at $r$''.  The first inequality holds since any $n_i$ is not $3$ and thus the induction hypothesis can be applied.  \nThe last inequality holds by $n\\ge 5$.  \n\nFor case (2), we further divide into two cases: (2.i), for every $i$ such that $n_i=3$, $T_i$ is Type B, and (2.ii) otherwise. For case (2.i), since an AK-rooted subgraph of $T_i$ of Type B in Figure \\ref{tree1} is only $T_i$ itself, the number is $1\\le 2^{3\/2}-1$. Thus, the similar analysis of Case (1) can be applied as follows: \n\\begin{align*}\n\\prod_{i:n_i\\neq 1, 3}(R(n_i)+1) \\cdot \\prod_{i: T_i \\mathrm{{\\ is\\ Type\\ B }}  }(2^{3\/2}-1+1) \\le \\prod_{i:n_i>1} 2^{n_i\/2}\\le 2^{n\/2}-1,     \n\\end{align*}\nthat is, the claim holds also in case (2.i). \n\nFinally, we consider case (2.ii). By the assumption, at least one $T_i$ is Type A in Figure \\ref{tree1}. Suppose that $T$ has $q$ children of $r$ forming Type A, which are renamed $T_1,\\ldots,T_q$ as canonicalization. Such renaming is allowed because we count non-isomorphic subtrees.  \nFurthermore, we can sort AK-rooted subtrees of $T_1,\\ldots, T_q$ as canonicalization. Since each Type A tree can form in $T'$ empty, a single vertex, or Type A tree itself, $T_1,\\ldots,T_q$ of $T$, the number of possible forms of subforests of $T_1,\\ldots,T_q$ of $T$ is \n\\[\n\\multiset{q}{3}=\\binom{q+2}{2}. \n\\]\nSince the number of subforests of $T_i$'s other than $T_1,\\ldots,T_q$ are similar evaluated as above, we can bound the number of AK-rooted subtrees by \\[\n\\binom{q+2}{2} \\prod_{i:i>q} 2^{n_i\/2} \\le \\frac{(q+2)(q+1)}{2} 2^{\\sum_{{i:i>q}} n_i\/2} \\le \\frac{(q+2)(q+1)}{2} 2^{(n-3q-1)\/2}.  \n\\]\nThus, to prove the claim, it is sufficient to show that $(q+2)(q+1)2^{(n-3q-3)\/2}\\le 2^{n\/2}-1$ for any pair of integers $n$ and $q$ satisfying $n\\ge 5$ and $1\\le q\\le (n-1)\/3$.  \nThis inequality is transformed to the following\n\\[\n\\frac{(q+1)(q+2)}{2^{\\frac{3(q+1)}{2}}} \\le 1-\\frac{1}{2^{\\frac{n}{2}}}.\n\\]\nSince the left hand and right hand of the inequality are monotonically decreasing with respect to $q$ and monotonically increasing with respect to $n$, respectively, the inequality always holds if it is true for $n=5$ and $q=1$. In fact, we have \n\\[\n\\frac{(1+1)(1+2)}{2^{\\frac{3(1+1)}{2}}}= \\frac{3}{4}=1 - \\frac{1}{2^2} \\le 1-\\frac{1}{2^{\\frac{5}{2}}}, \n\\]\nwhich completes the proof. \n\\end{proof}\n\n\\begin{theorem}\nThe winner of Arc Kayles on a tree with $n$ vertices can be determined in time $O^*(2^{n\/2})=O(1.4143^n)$.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{comment}\nEach type A has three position, no edge is selected, upper edge is selected and lower edge is selected.\nSo, this tree has rooted K-set at most\n\\begin{align*}\nT(n)\n&\\leq \\binom{a+2}{2}(T(n_1)+1)(T(n_2)+1)\\cdots(T(n_k)+1)\\\\\n&\\leq\\frac{1}{2}(a+1)(a+2)\\cdot2^\\frac{n_1}{2}\\cdot2^\\frac{n_2}{2}\\cdots2^\\frac{n_k}{2}\\\\\n&=\\frac{1}{2}(a+1)(a+2)\\cdot2^\\frac{n_1+n_2+\\cdots+n_k}{2}\\\\\n&=\\frac{1}{2}(a+1)(a+2)\\cdot2^\\frac{n-3a-1}{2}.\n\\end{align*}\nThen,\n\\begin{align}\n2^\\frac{n}{2}-1-T(n)\n&\\geq2^\\frac{n}{2}-1-\\frac{1}{2}(a+1)(a+2)\\cdot2^\\frac{n-3a-1}{2}\\notag\\\\\n&=2^\\frac{n}{2}\\left\\{1-\\frac{(a+1)(a+2)}{2^\\frac{3a+3}{2}}\\right\\}-1. \\label{eq1}\n\\end{align}\n\n\nWe consider function $f(a)=(2^\\frac{5}{2}-1)2^\\frac{3a+3}{2}-2^\\frac{5}{2}(a+1)(a+2)$.\nWhen $a$ is non-negative integer, $f(a)>0$ and\n\\begin{align}\n(2^\\frac{5}{2}-1)2^\\frac{3a+3}{2}-2^\\frac{5}{2}(a+1)(a+2)&>0\\notag\\\\\n(2^\\frac{5}{2}-1)2^\\frac{3a+3}{2}&>2^\\frac{5}{2}(a+1)(a+2)\\notag\\\\\n1-\\frac{(a+1)(a+2)}{2^\\frac{3a+3}{2}}&>\\frac{1}{2^\\frac{5}{2}}\\label{eq2}.\n\\end{align}\nFrom the formula\\ref{eq1} and formula\\ref{eq2},\n\\begin{align*}\n2^\\frac{n}{2}-1-T(n)\n&\\geq2^\\frac{n}{2}\\left(1-\\frac{(a+1)(a+2)}{2^\\frac{3a+3}{2}}\\right)-1\\\\\n&>2^\\frac{n}{2}\\cdot\\frac{1}{2^\\frac{5}{2}}-1\\\\\n&\\geq2^\\frac{5}{2}\\cdot\\frac{1}{2^\\frac{5}{2}}-1\\\\\n&>0.\n\\end{align*}\nTherefore, $n\\geq4$, $T(n)\\leq2^\\frac{n}{2}-1=O(1.4143^n)$\n\\end{comment}\n\n\\bigskip\n\nIn the rest of this section, we mention that we can determine the winner of Node Kayles for a tree in the same running time as Arc Kayles. The outline of the proof is also almost the same as Arc Kayles. Only the difference is to utilize the notion of \\emph{NK-rooted subtree} instead of \nAK-rooted subtree for Arc Kayles. \nFor $T=(V,E)$ rooted at $r$, a connected subtree $T'$ rooted at $r$ is called an NK-rooted subtree of $T$, if there exists an independent set $U  \\subseteq V$ such that $T[V\\setminus N[U]]=T'$.\n\n\n\n\n\\begin{lemma}\nAny tree rooted at $r$ has $O^*(2^{n\/2})(=O(1.4143^n))$ non-isomorphic NK-rooted subtrees rooted at $r$.  \n\\end{lemma}\n\n\\begin{proof}\nLet $\\hat{R}(n)$ be the maximum number of non-isomorphic NK-rooted subtrees of any tree rooted at some $r$ with $n$ vertices. \nSimilarly to Arc Kayles, we can show that $\\hat{R}(n)\\le 2^{n\/2}-1$ for all $n\\geq 4$ by induction. For $n\\le 4$, it is easy to see that the values of $\\hat{R}(n)$'s coincide with those of $R(n)$'s: $\\hat{R}(1)=R(1)=1, \\hat{R}(2)=R(2)=1, \\hat{R}(3)=R(3)=2$ (see Figure \\ref{tree1}), and $\\hat{R}(4)=R(4)=3$ (see Figure \\ref{tree2}). That is, $\\hat{R}(n)\\le 2^{n\/2}-1$ does not hold for $n=1$ and $3$, whereas it holds for $n=2$ and $4$, which shows the base step of the induction. We then consider the induction step. \n\nIn the induction step, we again take the same strategy as Arc Kayles; \nwe take a combination of the number of NK-rooted subtrees of $T_i$'s, which gives an upper bound on the number of NK-rooted subtrees of $T$. Since all the arguments use the same induction hypothesis and the same values of $\\hat{R}(n)=R(n)$ for $n=1,2,3,4$, the derived bound is the same as Arc Kayles. \n\\end{proof}\n\n\n\n\n\n\n\n\n    \n       \n       \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{FPT algorithm parameterized by vertex cover}\\label{sec:vc}\nIn this section, we propose winner determination algorithms for Colored Arc Kayles parameterized by the vertex cover number. As mentioned in Introduction, the selected edges in a play of Colored Arc Kayles form a matching. This implies that the number of turns is bounded above by the maximum matching size of $G$ and thus by the vertex cover number.  \nFurthermore, the vertex cover number of the input graph is bounded by twice of the number of turns of Arc Kayles. \nIntuitively, we may consider that a game taking longer turns is harder to analyze than games taking shorter turns. In that sense, the parameterization by the vertex cover number is quite natural. \n\n\nIn this section, we propose an $O^*(1.4143^{{\\tau}^2+3.17{\\tau}})$-time algorithm for Colored Arc Kayles, where ${\\tau}$ is the vertex cover number of the input graph.\nIt utilizes similar recursive relations shown in the previous section, \nbut we avoid to enumerate all possible positions by utilizing equivalence classification.\n\nBefore explaining the equivalence classification, we give a simple observation based on isomorphism. The isomorphism on edge-colored graphs is defined as follows. \n\n\\begin{definition}\\label{def:isomorphic:colored}\nLet $G^{(1)}=(V^{(1)},E_{\\CG}^{(1)}\\cup E_{\\CB}^{(1)}\\cup E_{\\CW}^{(1)})$ and $G^{(2)}=(V^{(2)},E_{\\CG}^{(2)}\\cup E_{\\CB}^{(2)}\\cup E_{\\CW}^{(2)})$ be edge-colored graphs. \nThen\n$G^{(1)}$ and $G^{(2)}$ are called \\emph{isomorphic} if for any pair of $u,v\\in V$ there is a bijection $f: V^{(1)}\\to V^{(2)}$ such that (i) $\\{u,v\\}\\in E_{\\CG}^{(1)}$ if and only if $\\{f(u),f(v)\\}\\in E_{\\CG}^{(2)}$, (ii) $\\{u,v\\}\\in E_{\\CB}^{(1)}$ if and only if $\\{f(u),f(v)\\}\\in E_{\\CB}^{(2)}$, and (iii) $\\{u,v\\}\\in E_{\\CW}^{(1)}$ if and only if $\\{f(u),f(v)\\}\\in E_{\\CW}^{(2)}$. \n\\end{definition}\n\nThe following proposition is obvious. \n\n\\begin{proposition}\\label{prop:isomorphic}\nIf edge-colored graphs $G^{(1)}$ and $G^{(2)}$ are isomorphic, \n$G^{(1)}$ and $G^{(2)}$ have the same game value for Colored Arc Kayles.\n\\end{proposition}\n\n\\begin{comment}\n\\begin{definition}\\label{def:isomorphic}\nLet $G=(V,E)$ and $G'=(V',E')$ be graphs. Then $G$ is \\emph{isomorphic} to $G'$ if for ant pair of $u,v\\in V$ there is a bijection $f: V\\to V'$ such that $\\{u,v\\}\\in E$ if and only if $\\{f(u),f(v)\\}\\in E'$. \n\\end{definition}\n\nWe further define isomorphism for colored graphs.\n\n\\begin{definition}\\label{def:isomorphic:colored}\nLet $G=(V,E)$ and $G'=(V',E')$ be edge-colored graphs where $E=\\bigcup_{i=1}^r E_i$ and $E'=\\bigcup_{i=1}^r E'_i$. Then $G$ is isomorphic to $G'$ if for ant pair of $u,v\\in V$ there is a bijection $f: V\\to V'$ such that $\\{u,v\\}\\in E_i$ if and only if $\\{f(u),f(v)\\}\\in E'_i$. \n\\end{definition}\n\nOur algorithm is also based on dynamic programming. However, we do not compute the winner on $G$ if the winner of $G'$ isomorphic to $G$ has already been computed. Instead, we use the result of $G'$.\nTo do this, we utilize memoization and preserve the result of $G$ if the winner of a graph isomorphic to $G$ is not computed.\n\n\n\\end{comment}\n\nLet $S$ be a vertex cover of $G=(V,E_{\\CG}\\cup E_{\\CW}\\cup E_{\\CB})$, that is, any $e=\\{u,v\\}\\in E_{\\CG} \\cup E_{\\CW} \\cup E_{\\CB}$ satisfies that $\\{u,v\\}\\cap S\\neq \\emptyset$. \nNote that for $v\\in V\\setminus S$, $N(v)\\subseteq S$ holds. \nWe say that two vertices $v,v'\\in V\\setminus S$ are \\emph{equivalent with respect to $S$ in $G$} if $N(v)=N(v')$ and $c(\\{u,v\\})=c(\\{u,v'\\})$ holds for $\\forall u\\in N(v)$. If two vertices $v,v'\\in V\\setminus S$ are equivalent with respect to $S$ in $G$, $G-u-v$ and $G-u-v'$ are isomorphic because the bijective function swapping only $v$ and $v'$ satisfies the isomorphic condition. Thus, we have the following lemma.\n\\begin{comment}\n\\begin{lemma}\\label{lem:isomorphic:type}\nFor $v,v'\\in V$ of the same type, the winner of Kayles on $G-u-v$ is the same as the one on $G-u-v'$ where $u\\in N(v)$.\n\\end{lemma}\n\\end{comment}\n\\begin{lemma}\\label{lem:isomorphic:type}\nSuppose that two vertices $v,v'\\in V\\setminus S$ are equivalent with respect to $S$ in $G$. Then, for any $u\\in N(v)$, $G-u-v$ and $G-u-v'$ have the same game value.\n\\end{lemma}\n\nBy the equivalence with respect to $S$, we can split $V\\setminus S$ into equivalence classes. Note here that the number of equivalence classes is at most $4^{\\card{S}}$, because for each $u\\in S$ and $v \\in V\\setminus S$, edge $\\{u,v\\}$ does not exist, or it can be colored with one of three colors if exists; \nwe can identify an equivalent class with ${\\bm{x}}\\in \\{\\emptyset,\\mathrm{G},\\mathrm{B},\\mathrm{W}\\}^{S}$, a 4-ary vector with length $\\card{S}$. For $S'\\subseteq S$, let ${\\bm{x}}[S']$ denotes\nthe vector by dropping the components of ${\\bm{x}}$ except the ones corresponding to $S'$. Also for $u\\in S$, ${\\bm{x}}[u]$ denotes the component  corresponding to $u$ in ${\\bm{x}}$. \nThen, $V$ is partitioned into $V_S^{({\\bm{x}})}$'s, where $V_S^{({\\bm{x}})}=\\{v\\in V\\setminus S \\mid \\forall u\\in S: c(\\{v,u\\})={\\bm{x}}[u]\\}$. We arbitrarily define  the representative of non-empty $V_S^{({\\bm{x}})}$ (e.g., the vertex with the smallest ID), which is denoted by $\\rho(V_S^{({\\bm{x}})})$. By using $\\rho$, we also define the representative edge set by\n\\begin{align*}\n    E^R(S)=\\bigcup_{{\\bm{x}}\\in \\{\\emptyset,\\mathrm{G},\\mathrm{B},\\mathrm{W}\\}^{S}} \\{\\{u,\\rho(V_S^{({\\bm{x}})})\\} \\in E_{\\CG}\\cup E_{\\CB} \\cup E_{\\CW} \\mid u \\in S\\}.\n\\end{align*}\nBy Lemma \\ref{lem:isomorphic:type}, we can assume that both players choose an edge only in $E^R(S)$, which enables to modify the recursive equations (\\ref{eq:recursion1}) and (\\ref{eq:recursion2}) as follows: \n\\begin{comment}\nBy Lemma \\ref{lem:isomorphic:type}, if there are at least two vertices with the same type, we only choose edges incident to one of them.\nThus, we define the representative edge set $E^R$ by\n\\begin{align*}\n    E^R=\\left\\{\\{u,v_i\\}\\in E \\mid \\nexists v_j (j<i) \\mbox{ such that $v_i$ and $v_j$ are equivalent.} \\right\\}.\n\\end{align*}\nWithout loss of generality, each player selects an edge from $E^R$.\n\n\\begin{lemma}\\label{lem:compute:representative}\nThe representative edge set $E^R$ can be computed in polynomial time.\n\\end{lemma}\n\\begin{proof}\nFor each $v\\in V$, we only check the neighborhoods of $v$. Thus, $E^R$ can be computed in polynomial time.\n\\end{proof}\nIn the step of the algorithm in Section \\ref{sec:Exalgo}, we compute the representative edge set $E^R$ and only select edges from $E^R$. That is, the recurrence formulas are changed as follows.\n\\end{comment}\nFor a vertex cover $S$ of $G$, we have  \n\\begin{align}\\label{eq:recursion3}\n    f_B(G) & = \\bigvee_{\\{u,v\\}\\in{(E_{\\CG}\\cup E_{\\CB})\\cap (E^R(S) \\cup S\\times S)}}\\lnot \\left(f_W(G-u-v))\\right),\\\\ \\label{eq:recursion4} \n    f_W(G) & = \\bigvee_{\\{u,v\\}\\in{(E_{\\CG}\\cup E_{\\CW})\\cap (E^R(S) \\cup S\\times S)}}\\lnot \\left(f_B(G-u-v))\\right).      \n\\end{align}\nNote that this recursive formulas imply that the winner of Colored Arc Kayles can be determined in time $O^*((\\tau^2 +\\tau\\cdot 4^{\\tau})^{\\tau})=\nO^*((4^{\\tau+\\log_4 \\tau})^{\\tau})=O^*(4^{\\tau^2+\\tau\\log_4 \\tau})$ $=O^*(5.6569^{\\tau^2})$, because the recursions are called at most $\\card{S}$ times and $\\tau+\\log_4 \\tau \\le 1.25 \\tau$ for $\\tau\\ge 1$. \n\n\\medskip \n\nIn the following, we give a better estimation of the number of induced subgraphs appearing in the recursion. Once such subgraphs are listed up, we can apply a standard dynamic programming to decide the necessary function values, or we can compute $f_B$ and $f_W$ according to the recursive formulas with memorization, by which we can skip redundant recursive calls. \nIn order to estimate the number of induced subgraphs appearing in the recursion, we focus on the fact that the position of a play in progress corresponds to the subgraph induced by a matching. \n \\begin{lemma}\\label{lem:positions:vc}\nThe number of nodes in recursion trees of equations (\\ref{eq:recursion3}) and (\\ref{eq:recursion4}) for Colored Arc Kayles is $O((r+1)^{\\card{S}^2\/4}3^{\\card{S}}\\card{S}^2)$, where $r$ is the used colors.\n \\end{lemma}\n\\begin{proof}\nSuppose that $S$ is a vertex cover of $G$ and players play Colored Arc Kayles on $G$. At some point, some edges selected by players together with their neighboring edges are removed, and the left subgraph represents a game position. Note that at least one endpoint of such a selected edge is in $S$, and selected edges form a matching. \nTo define such a subgraph, let us imagine that some matching $M$ is the set of edges that have been selected until the point. \nAlthough we do not specify $M$, the $M$ defines a partition $(X,Y,Z)$ of $S$; \n$X=\\{u\\in S \\mid  \\exists \\{u,v\\}\\in M: v\\in S\\}$,\n$Y=\\{u\\in S \\mid  \\exists \\{u,v\\}\\in M: v\\not\\in S\\}$,\nand $Z=S\\setminus (X\\cup Y)$. \nWe now count the number of positions having a common $(X,Y,Z)$. \nSince $X$ and $Y$ are removed, the remaining vertices in $V\\setminus S$ are classified into $V_Z^{({\\bm{x}})}$'s \n${\\bm{x}} \\in \\{\\emptyset,\\mathrm{G},\\mathrm{B},\\mathrm{W}\\}^{\\card{Z}}$.   \nThis is the common structure defined by $(X,Y,Z)$, and the positions vary as $\\card{Y}$ vertices in $\\bigcup_{{\\bm{x}} \\in \\{\\emptyset,\\mathrm{G},\\mathrm{B},\\mathrm{W}\\}^{\\card{Z}}} V_Z^{({\\bm{x}})}$ \nare matched with $Y$. Thus, we estimate the number of positions by counting the number of choices of $\\card{Y}$ vertices in $\\bigcup_{{\\bm{x}} \\in \\{\\emptyset,\\mathrm{G},\\mathrm{B},\\mathrm{W}\\}^{\\card{Z}}} V_Z^{({\\bm{x}})}$. Here, let $\\gamma$ be the number of used colors of edges. For example, BW-Arc Kayles and ordinary Arc Kayles use $\\gamma=2$ and $\\gamma=1$ colors, respectively, and which may reduce the numbers of ${\\bm{x}}$'s for smaller $\\gamma$. Then, it is above bounded\nby the multiset coefficient of $(\\gamma+1)^{\\card{Z}}$ multichoose $\\card{Y}$, \n\\[ \n\\multiset{(\\gamma+1)^{\\card{Z}}}{\\card{Y}}=\\binom{(\\gamma+1)^{\\card{Z}}+\\card{Y}-1}{\\card{Y}}\\le (\\gamma+1)^{\\card{Z}\\card{Y}}. \n\\]\nThis is an upper bound of the number of subgraphs to consider with respect to $(X,Y,Z)$. By considering all possible $(X,Y,Z)$, the total number of subgraphs is bounded by \n\\begin{align*}\n\\sum_{X,Y,Z\\subseteq S}(\\gamma+1)^{\\card{Z}\\card{Y}}      \n& \\le 3^{\\card{S}}\\cdot {\\card{S}}\\sum_{y=0}^{\\card{S}}(\\gamma+1)^{y({\\card{S}}-y)}\\le 3^{\\card{S}} \\cdot {\\card{S}}^2 \\cdot {(\\gamma+1)^{\\frac{{\\card{S}}^2}{4}}}, \n\\end{align*}\nwhere the first inequality comes from the choices of $X$, $Y$, and $Z$ from $S$. \n\\end{proof}\n\nThe following theorem immediately holds by Lemma \\ref{lem:positions:vc} and the fact that a minimum vertex cover of $G$ can be found in time $O^*(1.2738^{\\tau})$, where ${\\tau}$ is the vertex cover number of $G$~\\cite{chen2010improved}. \n\\begin{theorem}\\label{thm:algo:vc}\nThe winners of Colored Arc Kayles, BW-Arc Kayles, and Arc Kayles can be determined in time $O^*(1.4143^{{\\tau}^2+3.17{\\tau}})$, \n$O^*(1.3161^{{\\tau}^2+4{\\tau}})$, and  $O^*(1.1893^{{{\\tau}}^2+6.34{{\\tau}}})$, respectively, where ${\\tau}$ is the vertex cover number of a graph.  \n\\end{theorem}\n\n\n\n\\begin{comment}\n\\begin{align*}\n\\sum_{X,Y,Z\\subseteq S}\\left( (r+1)^{\\card{Z}}\\right)^{\\card{Y}}\n&=O(3^{{\\tau}}\\cdot {{\\tau}}\\sum_{y=0}^{{\\tau}} (r+1)^{y({{\\tau}}-y)})\\\\\n&=O(3^{{\\tau}} \\cdot {{\\tau}}^2 \\cdot {(r+1)^{\\frac{{{\\tau}}^2}{4}}})\\\\\n&=O(1.1893^{{{\\tau}}^2+6.34{{\\tau}}}\\cdot {{\\tau}}^2)\n\\end{align*}\n\\end{comment}\n\n\\begin{comment}\n\\begin{theorem}\\label{thm:algo:vc}\nGiven a graph $G$ of vertex cover number ${{\\tau}}$, ${\\bf Arc\\_Kayles\\_vc}(G)$ runs in time $O^*(2^{{{\\tau}}^2\/4}3^{{\\tau}})$.\n\\end{theorem}\n\\begin{proof}\nToDo\n\\end{proof}\n\n\\begin{corollary}\\label{cor:fpt:arc_kayles:vc}\nThe winner of Arc Kayles can be determined in time $O^*(1.1893^{{{\\tau}}^2+6.34{{\\tau}}})$\n\\end{corollary}\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\\if0\n\\subsection{Arc Kayles}\nIn this section, we propose an $O^*(1.1893^{{\\tau}^2+6.340{\\tau}})$-time algorithm for Arc Kayles, where $k$ is the vertex cover number of the input graph.\nIt utilizes the recursive relation shown in the previous section,\nbut we avoid enumerating all possible positions based on isomorphism. \nThis achieves the improvement of the running time for Arc Kayles.\n\n\n\n\n\n\n\\begin{definition}\\label{def:isomorphic}\nLet $G=(V,E)$ and $G'=(V',E')$ be graphs. Then $G$ is isomorphic to $G'$ if for ant pair of $u,v\\in V$ there is a bijection $f: V\\to V'$ such that $\\{u,v\\}\\in E$ if and only if $\\{f(u),f(v)\\}\\in E'$. \n\\end{definition}\n\n\nThen we can observe that the following key lemma holds.\n\\begin{lemma}\\label{lem:winner:isomorphic}\nIf $G$ is isomorphic to $G'$, then the winner of Arc Kayles on $G$ is the same as the one on $G'$.\n\\end{lemma}\n\nOur algorithm is also based on dynamic programming. However, we do not compute the winner on $G$ if the winner of $G'$ isomorphic to $G$ has already been computed. Instead, we use the result of $G'$.\nTo do this, we preserve the result of $G$ if the winner of a graph isomorphic to $G$ is not computed.\n\n\n\nIt is easily seen that twins $v,v'\\in V$, $G-u-v$ and $G-u-v'$ are isomorphic. Thus, we have the following lemma.\n\\begin{lemma}\\label{lem:isomorphic:twins}\nFor twins $v$ and $v'$, the winner of Keyles on $G-u-v$ is the same as the one on $G-u-v'$ where $u\\neq v,v'$.\n\\end{lemma}\n\n\n\n\nBy Lemma \\ref{lem:isomorphic:twins}, if there are twins, we only choose edges incident to one of them.\nThus, we define the representative edge set for $G$ by\n\\begin{align*}\nE^R(G)=\\left\\{\\{u,v_i\\}\\in E \\mid \\nexists v_j (j<i)  \\mbox{ such that $v_i$ and $v_j$ are twins.} \\right\\}.\n\\end{align*}\nWithout loss of generality, each player selects an edge from $E^R(G)$.\n\n\n\\begin{lemma}\\label{lem:compute:representative}\nGiven a vertex cover $S$ of $G$, the representative edge set $E^R(G)$ for $S$ can be computed in time $O(\\card{S}n)$.\n\\end{lemma}\n\\begin{proof}\nFor each $v\\in V$, we only check the neighborhoods of $v$. Since $S$ is a vertex cover, the degree of $v\\in V\\setminus S$ is at most $\\card{S}$. Thus, $E^R(G)$ can be computed in time $O(\\card{S}n)$.\\qed\n\\end{proof}\n\n\n\n\\begin{algorithm}[t]\n     \\caption{${\\bf Arc\\_Kayles\\_vc}(G)$, {\\bf Input:} a graph $G=(V,E)$}\n     \\begin{algorithmic}[1]\\label{keyles:vc}\n    \n    \\STATE Compute a minimum vertex cover $S$ of $G$\n    \\RETURN ${\\bf Recursive\\_Arc\\_Kayles\\_vc}(G,S)$\n \\end{algorithmic} \n \\end{algorithm}\n\n\n \\begin{algorithm}[t]\n    \\caption{${\\bf Recursive\\_Arc\\_Kayles\\_vc}(G,S)$, {\\bf Input:} a graph $G=(V,E)$, a vertex cover $S$}\n    \\begin{algorithmic}[1]\\label{keyles:recursive:vc}\n  \n    \\IF {$E=\\emptyset$}\n        \\RETURN $0$\n     \n    \\ELSIF {there is $G'$ isomorphic to $G$ that has already been computed}\n        \\RETURN ${\\bf Recursive\\_Arc\\_Kayles\\_vc}(G',S)$\n    \\ELSE\n         \\STATE Compute the representative edge set $E^R(G)$\n         \\RETURN $\\bigvee_{\\{u,v\\}\\in E(S)\\cup E^R}\\lnot \\left({\\bf Recursive\\_Arc\\_Kayles\\_vc}(G-\\Gamma(\\{u,v\\}),S)\\right)$\n     \\ENDIF\n \\end{algorithmic} \n \\end{algorithm}\n \n \n \\begin{lemma}\\label{lem:positions:vc}\nThe number of nodes in the recursive tree of ${\\bf Recursive\\_Arc\\_Kayles\\_vc}(G,S)$ is $O(2^{\\card{S}^2\/4}3^{\\card{S}})$.\n \\end{lemma}\n \\begin{proof}\n ToDo.\\qed\n \\end{proof}\n \n\\begin{theorem}\\label{thm:algo:vc}\nGiven a graph $G$ of vertex cover number ${{\\tau}}$, ${\\bf Arc\\_Kayles\\_vc}(G)$ runs in time $O^*(2^{{{\\tau}}^2\/4}3^{{\\tau}})$.\n\\end{theorem}\n\\begin{proof}\nWe can compute a minimum vertex cover $S$ of $G$ in time $O^*(2^{\\tau})$ by a standard branching algorithm.\n\nThe running time of ${\\bf Recursive\\_Arc\\_Kayles\\_vc}(G,S)$ can be bounded by the number of nodes of the recursive tree multiplied by the running time of the procedure in each node.\nIn Line 3, we have to check whether there is $G'$ isomorphic to $G$ has already been computed. To do this, we use memoization that preserves the information how vertices in vertex cover $S$ are connected to each other and how vertices in $V\\setminus S$ are connected to $S$ as a key. We only need $O(kn)$ bits to preserve a key and can detect \nThis can be bounded by $O^*(2^{{\\tau}^2\/4}3^{{\\tau}})$ by Lemmas \\ref{lem:compute:representative} and \\ref{lem:positions:vc}.\n\n\nThus, the total running time of ${\\bf Arc\\_Kayles\\_vc}(G)$ is $O^*(2^{{\\tau}})+O^*(2^{{{\\tau}}^2\/4}3^{{\\tau}})=O^*(2^{{{\\tau}}^2\/4}3^{{\\tau}})$.\\qed\n\\end{proof}\n\\fi\n\n\n\n\n\nWe have shown that the winner of Arc Kayles can be determined in time $O^*(1.1893^{{{\\tau}}^2+6.34{{\\tau}}})$. \nThe following theorem shows that the analysis is asymptotically tight,  \nwhich implies that for further improvement, we need additional techniques apart from ignoring vertex-cover-based isomorphic positions. We here give such an example in Figure \\ref{fig:lower}. \n\n\\begin{theorem}\\label{thm:lower:vc}\nThere is a graph for which the algorithm requires $2^{{\\tau}^2\/2}$ recursive calls for Colored Arc Kayles.\n\\end{theorem}\n\n\\begin{proof}\nIn this proof, we assume that $k$ is a multiple of $4$. \nWe explain how we systematically construct such a graph $G$ (see Figure \\ref{fig:lower}). \nLet $k$ be an even number. \nWe first define $U=\\{u_1,\\ldots, u_{k\/2}\\}$ and $V= \\{v_1, \\ldots, v_{k\/2}\\}$ as vertex sets. The union $U\\cup V$ will form a vertex cover after the graph $G$ are constructed.\nFor every 4-ary vector ${\\bm{x}}\\in \\{\\emptyset, \\mathrm{G},\\mathrm{B},\\mathrm{W}\\}^{U}$ and every $i\\in \\{1,\\ldots,k\/2\\}$, \nwe define $x_{i,{\\bm{x}}}$ as a vertex, and let $X$ be the collections of $x_{i,{\\bm{x}}}$'s, i.e., $X=\\{x_{i,{\\bm{x}}} \\mid {\\bm{x}}\\in \\{\\emptyset, \\mathrm{G},\\mathrm{B},\\mathrm{W}\\}^{U}, i\\in \\{1,\\ldots,k\/2\\}\\}$. These are the vertices of $G$. \nWe next define the set of edges of $G$. \nWe connect $v_i$ and $x_{i,{\\bm{x}}}$'s by black edges and by white edges for each $i\\in \\{1,2,\\ldots,k\/4\\}$ and $\\{k\/4+1,k\/4+2,\\ldots,k\/2\\}$, respectively.  \nFurthermore, we connect $x_{i,{\\bm{x}}}$ \nand $u\\in U$ \nfor each ${\\bm{x}}$ so that $\\{u,x_{i,{\\bm{x}}}\\}$ has \ncolor ${\\bm{x}}[u]$; if ${\\bm{x}}[u]=\\emptyset$, no edge exists between $x_{i,{\\bm{x}}}$ and $u\\in U$. Figure~\\ref{fig:lower} shows an example how we connect $x_{1,{\\bm{x}}}$ and $u_i$'s, where \n${\\bm{x}}[u_1]=\\emptyset$, ${\\bm{x}}[u_2]=\\mathrm{G}$, ${\\bm{x}}[u_i]=\\mathrm{W}$, and ${\\bm{x}}[u_{k\/2}]=\\mathrm{B}$. Notice that in Figure \\ref{fig:lower}, $\\{x_{1,{\\bm{x}}},u_1\\}$ is connected with edge $\\emptyset$ for explanation, which means that there is no edge between $x_{1,{\\bm{x}}}$ and $u_1$. \nNote that the number of vertices in $G$ is $\\card{U}+\\card{V}+\\card{X}=k+4^{k\/2}k\/2$. \nMoreover,\n$U$, $V$, and $X$ form independent sets and $X$ separates $U$ and $V$.\nThus, $U\\cup V$ forms a vertex cover of size $k$ in $G$.\n\n\nWe are ready to explain that $G$ has different $2^{{\\tau}^2\/2}$ subgraphs called by the algorithm. \nStarting from $G$, we call the recursive formulas (\\ref{eq:recursion3}) and (\\ref{eq:recursion4}) $k\/2$ times by selecting edges incident to only $v_i$'s. \nThen, all the vertices in $V$ are removed from $G$, and the neighbors of remaining $x_{i,{\\bm{x}}}$'s are in $U$. That is, each $x_{i,{\\bm{x}}}$ has its inherent set of neighbors, and thus \n$x_{i,{\\bm{x}}}$'s are not equivalent each other for vertex cover $U\\cup V$.\nThis implies that if the set of removed edges are different, the resulting subgraphs are also different. \n\nIn a step before $k\/2+1$, an edge connecting some $v_i$ is removed, and such an edge is chosen from $\\{\\{v_i,x_{i,{\\bm{x}}}\\}\\mid {\\bm{x}}\\in \\{\\emptyset, \\mathrm{G},\\mathrm{B},\\mathrm{W}\\}^{U}\\}$, that is, the number of candidates is $4^{k\/2}$ for each $i$. \nThus, the total way to choose edges is $(4^{k\/2})^{k\/2}=4^{k^2\/4}=2^{k^2\/2}$; at least $2^{k^2\/2}$ recursion calls occur.  \n\\end{proof}\n\n\\begin{figure}[htbp]\n \\centering\n \\includegraphics[width=0.8\\linewidth]{vc_lower.pdf}\n \\caption{The constructed graph $G=(U\\cup V\\cup X, E)$.}\n \\label{fig:lower}\n\\end{figure}\n\nBy the similar construction, we can show the following theorem.\n\n\\begin{theorem}\\label{thm:lower:vc2}\nThere is a graph for which the algorithm requires $1.3161^{{\\tau}^2}$ and $1.1893^{{\\tau}^2}$ recursive calls for BW-Arc Kayles and Arc Kayles, respectively.\n\\end{theorem}\n\n\n\\begin{remark}\nAlthough Theorems \\ref{thm:lower:vc} and \\ref{thm:lower:vc2} give lower bounds on the running time of the vertex cover-based algorithms, the proof implies a stronger result. In the proof of Theorem \\ref{thm:lower:vc}, we use ID's of the vertices in $U$. By connecting $i$ pendant vertices to $u_i$, we can regard them as ID of $u_i$. Furthermore, we make $U$ a clique by adding edges. These make the graphs not automorphic, which implies that the time complexity of an algorithm utilizing only isomorphism is at least the value shown in Theorems \\ref{thm:lower:vc} or \\ref{thm:lower:vc2}.    \\end{remark}\n\n\n\n\n\n\n\n\n\n\n \n \n \n    \n\n  \n     \n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection*{Acknowledgments}\nThis work is partially supported by JSPS KAKENHI JP20H05967, JP21H05852, JP21K17707, JP21K19765, JP21K21283, JP22H00513.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWhen observing and characterising the Universe on large scales, there are two broadly different, yet intertwined, types of observations \\citep{1980lssu.book.....P}. In the first type of observation, one endeavours to constrain the expansion rate of the Universe at different times. This can be done by measuring the expansion rate itself or through a variety of cosmological distance measures: angular diameter distances, luminosity distances, standard sirens, etc. In particular, if $a(t)$ is the scale factor of the Universe at cosmic time $t$, the expansion rate at that time can be defined as $H[a(t)]={\\dot a}\/{a}$ (where the overdot is derivative with regards to cosmic time). Then, one can derive an expression for the radial comoving distance, $D_M$, by following a radial null-geodesic.\n\\begin{equation} \\label{eq:distance}\n    D_M \\equiv \\chi = \\int_t^{t_0}\\frac{dt'}{a(t')}=\\int_0^{z}\\frac{dz'}{H(z')}\n    \\, ,\n\\end{equation}\nwhere we use unit $c=1$ throughout, $t_0$ is cosmic time today and $z$ is the cosmological redshift, $1+z=1\/a$. Moreover, assuming zero spatial curvature, it is possible to derive expressions for the luminosity ($D_L$) and angular diameter distances ($D_A$) by multiplying and dividing Eqn. \\ref{eq:distance} by $(1+z)$, respectively:\n\\begin{equation} \\label{eq:distances}\n    D_L = (1+z) D_M \\quad \\text{and} \\quad\n    D_A = D_M\/(1+z).\n\\end{equation}\nThus, measurements of $D_L$ and $D_A$ at different redshifts can be used to reconstruct $H(z)$ over time.\n\nIn the second type of observation, one focuses on the rate with which structures undergo gravitational collapse on the expanding background. Under the assumptions that the bulk of matter in the recent past can be described by a presureless fluid and that neutrino effects can be neglected, the density contrast, $\\delta\\equiv\\rho\/{\\bar \\rho}-1$ (where ${\\bar \\rho}$ is the mean matter density of the Universe), obeys an evolution equation of the form:\n\\begin{eqnarray}\nf'+f^2+\\left(1+\\frac{{\\rm d}\\ln aH}{{\\rm d}\\ln a}\\right)f=\\frac{3}{2}\\Omega_m(z) \\, ,  \\label{eq:growth}\n\\end{eqnarray}\nwhere the prime denotes a derivative with regards to $\\ln a$, the growth rate is defined as $f\\equiv {\\rm d}\\ln \\delta\/{\\rm d}\\ln a$ and $\\Omega_m(z)$ is the fractional energy density in matter as a function of redshift. The latter quantity depends, through the Einstein field equations, on $H \\equiv H(z)$ so that\n\\begin{eqnarray} \\label{eq:jeans}    \n\\Omega_m(z)=\\frac {\\Omega_m H^2_0}{a^3H^2},\n\\end{eqnarray}\nwith $\\Omega_m \\equiv \\Omega_m(0)$, which we use for ease of notation.\nThus, a measurements of the growth rate of structure at different redshifts (or times) can also be used to reconstruct $H(z)$ over time as well as the fractional matter density today, $\\Omega_m$.\n\nThe challenge of modern cosmology has been to use a wide range of different cosmological observables to constrain $H(z)$ as a function of time or redshift (and, of course, $\\Omega_m$) and, crucially, to pin down the underlying cosmological model which describes $H(z)$ in terms of a greatly reduced set of cosmological parameters. The frontrunner is the $\\Lambda$ Cold Dark Matter ($\\Lambda$CDM) model, whose parameters are now constrained to an unprecedented precision \\citep{Planck}. However, interestingly, inconsistencies, or \"tensions\", have begun to emerge. Different probes are leading to different constraints on, for example, the Hubble constant, $H_0$ \\citep[e.g.][]{riess_42, TRGB_H0, Maser_H0} or the density-weighted amplitude of fluctuations, $S_8 \\equiv \\sigma_8 \\left(\\Omega_m\/0.3\\right)^{0.5}$ \\citep[e.g.][]{2007.15633, 2105.13549, 1607.00008, 2106.01208, K1K, 2105.09545, Garcia-Garcia21, 2021arXiv211109898W}, where $\\sigma_8$ is the variance of $\\delta$, in spheres of radius $8\\,h^{-1}$Mpc.\n\nIn this paper we ask if it is possible to obtain meaningful, or precise, constraints on cosmological parameters with minimal assumptions about the cosmological model. To be more specific, we step back and try to find model-independent constraints on $H(z)$ and $\\Omega_m$  from measurements of the expansion history itself, cosmological distances and the growth rate. This allows us to contrast the constraining power of these two very different sets  of observables, to explore how combining them improves constraints and, most importantly, how much constraints improve once one assumes a cosmological model.\nThe hope is that understanding this process will shed light on the analysis of theories that go beyond  $\\Lambda$CDM, but also may have bearing on the current inconsistencies in parameter constraints. We will implement our model-independent approach using Gaussian processes \\citet{gp_book, Seikel_12}. Gaussian processes allow us to  reconstruct a well-defined distribution of histories of $H(z)$. The data then allows us to constrain the parameters of this distribution and, in doing so, tells us at what redshifts $H(z)$ is well determined and at what redshifts it is determined poorly.\n\nThe literature already hosts a number of examples of the possible uses of Gaussian processes to test certain aspects of the present cosmological paradigm in model-agnostic ways. For example,  one can find tests for the model dependence of the $H_0$ \\citep{1802.01505, 1907.10813, Liao_H_gp, Bonilla_H0_gp} and $S_8$ \\citep{S8_gp} tensions, for the non-zero curvature of space-time \\citep{2011.11645, 2007.05714}, as well as tests for the density and the equation of state of dark energy \\citep{1902.0942, 1806.02981}.  More closely related to the topic of this work, in \\citet{2105.01613} Gaussian processes were used to study the statistical correlations between the expansion history, cosmological distances, and the linear growth rate without appealing to the physical relationships between the three functions. However, the clearest precedent of the methodology used here is \\citet{1911.12076} who already employ Gaussian processes to obtain model-independent constraints for $\\Omega_m$ and $\\sigma_{8}$ based on the relationship between the expansion history and the linear growth rate.  The methodology developed in the present paper expands and improves many aspects of their analysis. First, our Gaussian process extends all the way to recombination. This allows us to solve the Jeans equation (Eqn. \\ref{eq:growth}) without using fitting formulas. Moreover, we also employ cosmological distances to constrain the evolution of the expansion history. This, in combination with the extended range of the Gaussian process, allows us to use the position of the first acoustic peak of the CMB temperature power spectrum to constrain expansion history far into the past. Finally, and most importantly, we sample our Gaussian process simultaneously with the cosmological parameters, allowing us to observe potential correlations between the two.\n\nThe structure of this paper is as follows: in Section~\\ref{Sect: Methodology} we present the methodology of this work regarding the use of Gaussian Process to compute predictions for cosmological observables. In Section~\\ref{Sect: Observables and data sets} we describe the cosmological observables from which we employ data and motivate their use in the context of this work. We present our results in Section~\\ref{Sect: Results} and discuss the implications of our work in Section ~\\ref{Sect: Conclusions}.\n\n\\section{Methodology} \\label{Sect: Methodology}\n\nThroughout this paper we will be working with the Eqns. \\ref{eq:distance} and \\ref{eq:growth} to connect the time evolution of $H(z)$ with current and future observations. In the case of direct measurements of $H(z)$, the expansion history can be trivially matched to observations without the need to apply any transformation.  In the case of distances, we can see from Eqn. \\ref{eq:distance} that it is possible to generate predictions for the observables from any choice of $H(z)$ by performing the relevant integral. In the case of growth (Eqn. \\ref{eq:growth}), we are faced with the problem that most observations do not report $f$ but the combined quantity $ f\\!\\sigma_{8}$. Thus, it is convenient to rewrite Eqn. \\ref{eq:growth} in terms of the latter. This can be done as follows: assuming that perturbations grow in a self-similar manner, we can  define $\\delta(t,{\\bf x})\\equiv D(t)\\delta_0({\\bf x})$, where $\\delta_0$ is the density contrast today. Then, we can rewrite Eqn. \\ref{eq:growth} as\n\\begin{equation} \\label{eq:jeans_D}\n    \\frac{d}{da}\\left(a^2H\\frac{dD}{d\\log a}\\right)=\\frac{3}{2}\\Omega_m(a)\\,a\\,H\\,D \\, .\n\\end{equation}\nDefining $y\\equiv a^2E\\frac{dD}{d\\log a}$, with $E\\equiv H\/H_0$, and switching to the integration variable $s\\equiv\\log(1+z)$, this equation can be written in terms of a system of coupled  first-order equations:\n\n\\begin{equation} \\label{eq:Jeans_sys_D}\n    \\frac{dy}{ds}=-\\frac{3}{2}\\frac{\\Omega_m}{a E}D \\, ,\\hspace{12pt} \n    \\frac{dD}{ds}=-\\frac{y}{a^2E}.\n\\end{equation}\n\nThen, it is possible to transform the quantities $y(s)$ and $D(s)$ into $f\\!\\sigma_{8}(s)$ and $\\sigma_{8}(s)$ by applying the following transformation\n\\begin{equation} \\label{eq:D_to_s8}\n    \\sigma_{8}(s) = \\sigma_{8}(0) \\frac{D(s)}{D(0)}\\,,\\hspace{12pt}\n    f\\!\\sigma_{8}(s) = \\frac{y(s) \\, \\sigma_{8}(0)}{E(s)D(0)}e^{2s}\\, .\n\\end{equation}\nThe thrust of this paper is to keep the analysis as model-independent as possible. Yet, as we can see, it is still possible to extract information about some of the cosmological paramaters. For a start, combining the information from distances and growth allows us to constrain $\\Omega_m$. But we also have, automatically, $H_0\\equiv H(z=0)$ and, as we just saw, we can calculate $\\sigma_8\\equiv\\sigma_8(z=0)$ (or $S_8$ as a derived parameter).\n\nIn this paper we will model the time (or redshift) dependent Hubble rate, $H(z)$, as a Gaussian Process spanning over the redshift range $0<z<1100$. A Gaussian process (GP) is a collection of random variables, any finite number of which have a joint Gaussian distribution \\citep{gp_book}. A GP $g(\\textbf{x})$, where $\\boldsymbol{x}$ is a collection of random variables, is fully specified by a mean function $m(\\textbf{x}) \\equiv \\mathcal{E}[g(\\textbf{x})]$, where $\\mathcal{E}[\\cdots]$ is the expectation value over the ensemble,  and a covariance function $k(\\textbf{x}, \\textbf{x'}) \\equiv \\mathcal{E}[(g(\\textbf{x})-m(\\textbf{x}))(g(\\textbf{x'})-m(\\textbf{x'}))^T]$. In combination, the mean and covariance functions determine the statistical properties of the ensemble of random variables which defines the family of shapes that the GP can take. \n\nThus, GP's can be used as agnostic function-space priors which, in combination with a likelihood for the observed data $\\mathcal{L}(\\textbf{y}| g(\\textbf{x}), \\boldsymbol{\\sigma})$, where $\\textbf{y}$ is a set of data points with a set of errors $\\boldsymbol{\\sigma}$),  define a regression model. Then, observations can be used to inform the GP posterior (i.e. the statistical properties of the set of random variables) $\\mathcal{P}(g(\\textbf{x})| \\textbf{y},\\boldsymbol{\\sigma})$, which determines the subset of functions most consistent with the data.\n\n Since the Hubble rate is generally regarded as a monotonically increasing function, it is important to define a non-zero mean for the GP. This prevents the GP from simply fitting the long range upwards trend of $H(z)$ while washing out interesting local features \\citep{Shafieloo_gp}. In this paper, we will define the mean of the GP in terms of a fiducial $\\Lambda$CDM prediction:\n\\begin{equation} \\label{H_bar}\n    \\overline{H}(z) \\equiv  \\overline{H_0} \\sqrt{\\overline{\\Omega}_m(1+z)^3+\\overline{\\Omega}_r(1+z)^4+ \\overline{\\Omega}_{\\Lambda}} \\, ,\n\\end{equation}\nwhere $\\overline{\\Omega}_m$,  $\\overline{\\Omega}_r$ and $\\overline{ \\Omega}_{\\Lambda}$ stand for the present values of the cosmological densities of matter, radiation and dark energy , respectively,\nas given by the \\textit{Planck} 2018 TTTEEE+lowE cosmology (see Tab. \\ref{tab:fiducial}). In other words, they are completely fixed and are not variables.\n\nIn order to prevent this (ultimately arbitrary) choice of mean from biasing our constraints on the cosmological parameters, we define a free amplitude parameter $A_0$ that multiplies the mean of the GP such that\n\\begin{equation} \\label{H_mean}\n    H_m(z) \\equiv A_0 \\, \\overline{H}(z) \\, .\n\\end{equation}\nSimply put, if the data is somewhat away from what one might expect from the fiducial $\\Lambda$CDM background, the Gaussian process will have to soak up the large scale differences, or trends, in detriment to the small scale, local features. Moreover, in those regions where data is sparse, the GPs constraints are highly dominated by their prior; i.e. the chosen mean. If such mean systematically falls beneath or above the data, it can lead to spurious trends in the final results. In App. \\ref{app:validation}, we show these effects can substantially bias the constraints we obtain in the absence of $A_0$. For these reasons, we advise against common practices such as employing a constant as a mean for a GP modelling the recent expansion history or extrapolating GP results to regions where they become dominated by the choice of mean and covariance function. \n\n\\begin{table} \n    \\caption{ Values of the cosmological parameters used to define the fiducial cosmology. Values based of \\textit{Planck} 2018 TTTEEE+lowE cosmology \\citep{Planck}}\n    \\centering\n    \\begin{tabular}{|p{1.5 cm}|p{1.5cm}}\n    \\hline\n    Parameter & value   \\\\\n    \\hline\n    $\\overline{\\Omega}_{\\rm{m}}$   &  0.316  \\\\\n    $\\overline{\\Omega}_{\\Lambda}$ & 0.683 \\\\\n    $\\overline{\\Omega}_r$ &  $9.245 10^{-5}$ \\\\\n    $\\overline{H}_0$ & 67.27 \\\\\n\\end{tabular}\n\\centering\n\\label{tab:fiducial}\n\\end{table}\n\nThe Hubble rate must also be a continuous and smooth function. Therefore it is important that the covariance function of the GP reflect such properties \\citep{Dialektopoulos21}. Ultimately, we chose a a square exponential covariance function to model the correlations between the different nodes of the GP. This decision was made based on the fact that the square exponential is computationally inexpensive and infinitely differentiable kernel appropriate for modelling smooth fluctuations around the mean of the GP. Mathematically, the square exponential covariance function is defined as\n\\begin{eqnarray} \nk\\left[g(\\boldsymbol{x}),g(\\boldsymbol{x}')\\right]=\\eta^2\\exp\\left[\\frac{-|g(\\boldsymbol{x})-g(\\boldsymbol{x}')|^2}{2l^2} \\right] \\, ,  \\label{eq:QuadExp}\n\\end{eqnarray}\nwhere $\\eta$ is the amplitude of the oscillations around the mean and $l$ is the correlation length between the GP nodes. The parameters $\\eta$ and $l$ are known as hyperparameters since they constrain the possible values that the nodes; i.e. the parameters the GP, can take. In other words, the hyperparameters dictate the family of functions available to the GP.  In addition to this, a white noise term with amplitude $10^{-3}$ was added to the covariance function to ensure numerical stability.   \n\nSampling over GPs is extremely efficient when the model for the observed data is linearly related to the GP and the likelihood is Gaussian. In this case, it can be shown that the posterior distribution also takes the form of a GP for which the mean and covariance functions can be calculated analytically by marginalizing over the nodes of the original GP \\citep{gp_book}. However, when the observed data is not linearly related to the GP, the posterior distribution becomes non-Gaussian. Thus, each node of the GP; i.e. each dimension of the multivariate Gaussian distribution, must be treated as a newly added degree of freedom in the sampling process (i.e. a new parameter). \n \n\\begin{figure*} \n\\centering\n    \\includegraphics[width=.65\\linewidth]{{\"schematic\"}.pdf} \n    \\caption{Shows a schematic representation of the data analysis pipeline.}\n    \\label{fig:code}\n\\end{figure*} \n\nExploring such a vast parameter space is effectively unfeasible with traditional non-gradient-based sampling algorithms and requires more sophisticated samplers. In this work we employ the No U-turn Sampler (NUTS) \\citep{NUTS}, a self-tuning variant of Hamiltonian Monte Carlo (HMC) \\citep{HMC, Betancourt17} which simulates Hamiltonian trajectories through the parameter space to generate efficient proposal steps. This allows us to handle hundreds or thousands of parameters during the inference process efficiently. \nWe employ the NUTS implementation of the python package \\texttt{Pymc3} \\citep{Pymc3} which makes use of the tensor package \\texttt{Theano} \\citep{Theano} as a back-end to build a symbolic graph of the model used to fit the data. This graph is then used to perform automatic-differentiation \\citep{Autodiff18} to obtain the gradient of the regression model needed for HMC. Through out the likelihood we make use of simple integration schemes; e.g. Runge-Kutta, trapezoidal rule, on which \\textsc{Theano} can apply automatic-differentiation to make the likelihood differentiable in all parameters.\n\nIn the case of distances (see Eqn. \\ref{eq:distance}), the integral over $1\/H_{\\mathcal{GP}}$ was performed using the trapezoidal rule with $s\\equiv \\log(1+z)$ as the integration variable. In the case of the linear growth rate, the system of differential equations shown in Eqns. \\ref{eq:Jeans_sys_D} was solved for $y(s)$ and $D(s)$ employing a second order Runge-Kutta scheme, integrating over the redshift range $0 < z < 1100$ from the past into the present with initial conditions $D(z=1100)=a(z=1100)$ and $y(z=1100)=a(z=1100)^3E(z=1100)$. Note that these initial conditions are only strictly correct for a purely matter-dominated universe, which is not accurate at $z\\sim1000$. However, the impact of this assumption is negligible by comparison to current uncertainties on growth measurements. In combination, these methods allow us to obtain sub-percentage accuracy through out the entire redshift range of the GP with respect the output of cosmological CLASS \\citep{Class}, which performs the full numerical calculation.\n\nIn order to cover the redshift range $0 < z < 1100$ while keeping the numerical errors under control, we sampled the GP evenly in $s$ as opposed to redshift itself. This variable concentrates most of the GP samples at low redshift where most of our data lays while offering great numerical accuracy \\citep{1710.0423}. This allowed us to cover our desired redshift range with only $200$ nodes; i.e. with a 200-dimensional GP. \n\nIn addition to the $200$ parameters associated with the GP, we also sample over the cosmological parameters present in Eqn. \\ref{eq:growth}; i.e. $\\Omega_m$ and $\\sigma_{8}$, as well as the GP hyperparameters $\\eta$ and $l$ described in Eqn. \\ref{eq:QuadExp}. We define the GP not over the Hubble rate itself but over as relative deviations from the $\\Lambda$CDM background \n\\begin{eqnarray}\n  H_{\\mathcal{GP}} \\equiv H_m(s) [1 + \\delta H_{\\mathcal{GP}}(s)], \\nonumber\n\\end{eqnarray}\nwhere $\\delta H_{\\mathcal{GP}}$ is a GP centred at zero. This approach has the added benefit of normalizing the amplitude of the oscillations of the GP with respect the mean making the sampling of $\\eta$ far more efficient by virtue of reducing its potential range of values. Note that the correlation length scale $l$ is defined in units of $s$ as opposed to $z$. We also marginalize over the absolute magnitude of the supernovae $M$ (See Sect. \\ref{subsec: Supernovae} for details), the scale of the sound horizon $r_s$, and the amplitude of $H_m$, $A_0$.\n\nIt is important to emphasise that we perform a fully Bayesian inference over the GP, as opposed to what it is known as an empirical Bayesian analysis in which first, the marginal likelihood of the GP hyperparameters is maximized; and then, keeping their values fixed, the conditional posterior over the GP is inferred. While for large sets of data the output of the two approaches converges to the same results, when only sparse data is available the fact that the empirical Bayesian analysis does not account for the full posterior volume of the hyperparameters  can lead to an under-accounting of uncertainties.\n\nIn order to establish the reliability of the methodology developed in this work for cosmological inference, and the robustness of the fiducial results, we also considered a series of alternative analyses. In the first test, the hyperparameters of the GP ($\\eta$, $l$) were kept fixed. We will label results from this analysis as \"Fixed HP\". In the second test, instead of sampling over the $A_0$ parameter, we performed a two-step analysis. In the first step, a standard $\\Lambda$CDM model was fit to the data. Based on this best-fit $\\Lambda$CDM model, a expansion history was derived. Then, in the second step, the best-fit expansion history was used as the mean of the GP during the analysis while fixing $A_0 = 1$. We will label results from this analysis as \"Two-Steps\". \n\n\\begin{table} \n\\caption{Sampled parameters and their priors in the standard analysis (first column), the fixed hyperparameters analysis (second column) and the Two-Steps analysis (third column). }\n\\begin{tabular}{ |p{1cm}|p{1.75cm}|p{1.75cm}|p{1.75cm}|p{1.75cm}|p{1.75cm}}\n \\hline\n  Parameter & Fiducial & Fixed HP & Two-Steps  \\\\\n    \\hline\n    $\\eta$ & $\\mathcal{N}_{1\/2}(0,0.2)$  & 0.2 & $\\mathcal{N}_{1\/2}(0,0.2)$  \\\\\n    $l$ & $U(0.01, 6)$  & 1.0 & $U(0.01, 6)$   \\\\\n    \\hline\n    $\\Omega_m$ & $U(0,1)$ & $U(0,1)$ & $U(0,1)$ \\\\\n    $\\sigma_{8}$ & $\\mathcal{N}(0.8, 0.5)$ & $\\mathcal{N}(0.8, 0.5)$ & $\\mathcal{N}(0.8, 0.5)$   \\\\\n    \\hline\n    $A_0$ & $U(0.8, 1.2)$ & $U(0.8, 1.2)$ & 1 \\\\\n    $M$ & $\\mathcal{N}(-19.2, 1)$ & $\\mathcal{N}(-19.2, 1)$ & $\\mathcal{N}(-19.2, 1)$   \\\\\n    $r_s$ & $\\mathcal{N}(150, 5)$ & $\\mathcal{N}(150, 5)$ & $\\mathcal{N}(150, 5)$ \\\\\n\\end{tabular}\n\\label{tab:prior}\n\\end{table}\n\nA summary of the priors used in the fiducial and alternative analyses can be found in Tab. \\ref{tab:prior}. As a general rule, when performing Bayesian inference one should avoid broad, uniform priors \\citep{Gelman17}. This is mainly due to the fact that they do not accurately represent the prior knowledge, put a lot of posterior mass in unlikely values and introduce hard boundaries which can be difficult to motivate. This last consideration is particularly important when using \\textsc{HMC} since sharp edges in parameter space can lead to the sampling process becoming inefficient. Following this principle, we employed a half-Gaussian distribution with zero mean and standard deviation $0.2$ as prior distribution for the hyperparameter $\\eta$ to down-weight extreme deviations (i.e. $20\\%$ and above, well within the observational errors on $H(z)$) from the chosen GP mean without introducing any unnecessary hard boundaries. In the case of $l$ we used a uniform distribution between $0.01 < l < 6$ for the following reasons. First, there is no reason to down-weight long correlation modes against short ones or vice versa. Second, looking at Eqn. \\ref{eq:QuadExp}, it is possible to see that as $l \\rightarrow 0$ the value of the covariance function approaches zero regardless of the value of $\\eta$. This opens a vast a volume in the parameter space which can lead to an inefficient sampling. Third, the likelihood function becomes flat at such small scales since there is no information in the data to constrain those small scale modes. Forth, the expansion rate is expected to have some degree of temporal correlations, and therefore the limit $l=0$ must be excluded. The parameter $\\Omega_m$ was sampled from a uniform distribution between $0$ and $1$ to enforce the physical boundaries on the allowed values for the cosmological matter fraction. In the case of $\\sigma_8$ for which there is no physically-motivated upper limit, we employ a better behaved Gaussian prior centered at $0.8$ and with $0.5$ standard deviation. Given the degeneracy of $A_0$ with the GP parameters, the parameter was sampled from a uniform distribution between $0.8$ and $1.2$, which amply encompasses the current discrepancy in $H_0$ between CMB data and local measurements. The supernova absolute magnitude parameter $M$ was sampled from a Gaussian distribution centered at value found by the SH0ES collaboration for the parameter $M = -19.2$, with a standard deviation of $1$. While the mean value corresponds to a local expansion rate of $H_0 \\simeq 74.0$ km\/s\/Mpc, the standard deviation ensures that all values in the range $50 < H_0 < 100$ km\/s\/Mpc fall within the $1\\sigma$ region of the prior distribution. Finally, the sound horizon scale, $r_s$,  was sampled from a Gaussian distribution centered at $150$ Mpc with a standard deviation of $5$ Mpc. We do this instead of computing $r_s$ from the expansion history in combination with the BBN prior to allow for largest deviations from the fiducial cosmology. This was needed to recover the test cosmologies of App. \\ref{app:validation}.\n\n\n\\section{Observables and Data sets} \\label{Sect: Observables and data sets}\nIn order to make the most of the modelling freedom offered by GPs we consider as much data as possible. In this work we use a combination of several different probes that together account for 91 data points for a variety of cosmological observables. A summary of the data used in this work can be found in Tab. \\ref{tab:data}.\n\n\\begin{figure*} \n\\centering\n    \\includegraphics[width=\\linewidth]{{\"data\"}.pdf} \n    \\caption{Data points from the different surveys used in this work across redshift for the three cosmological functions of interest $H(z)$, $D_M$ and $f \\! \\sigma_8$.}\n    \\label{fig:data}\n\\end{figure*} \n\n\\begin{table*} \n\\centering\n\\caption{Data sets used in our analysis, listing the probe, the redshift range of the probe, the choice of observable and the size of the data vector. See Fig. \\ref{fig:data} for a pictorial representation. }\n\\begin{tabular}{|p{5.5cm}|p{2cm}|p{1.5cm}|p{0.7cm}|p{.7cm}|p{0.7cm}|p{1cm} |  }\n \\hline\nData set &  Probe & Redshifts &\\multicolumn{3}{|c|}{Observable} & Data Points\\\\\n&  & & $H(z)$ & $D_M(z)$ & $f \\! \\sigma_8 $ & \\\\\n \\hline\n CCs \\citep{2011.11645} &  Cosmic Chronometers & 0.07 - 2.36 & $\\checkmark$ & $\\times$ & $\\times$ & 35\\\\\n \nPantheon DS17\\citep{1710.00845} & SNe Ia & 0.38 - 0.61 & $\\times$ & $\\checkmark$ & $\\times$ & 40\\\\\n \n BOSS DR12 \\citep{1607.03155} & BAO+RSD & 0.38 - 0.61 & $\\checkmark$ & $\\checkmark$ & $\\checkmark$ & $3\\times3$\\\\\n\n eBOSS DR16 \\citep{2007.08991} & BAO+RSD & 1.48 & $\\checkmark$ & $\\times$ & $\\times$  & $1\\times3$ \\\\ \n \n Wigglez \\citep{1204.3674} & RSD & 0.44 - 0.73 & $\\times$ & $\\times$ & $\\checkmark$  & $1\\times3$ \\\\\n \n DSS \\citep{2105.05185} & RSD & 0 & $\\times$ & $\\times$ & $\\checkmark$  & 1 \\\\\n \n \\textit{Planck} 2018  \\citep{Planck} & CMB & 1090.30 & $\\times$ & $\\checkmark$ & $\\times$  & 1 \\\\\n \\hline\n DESI \\citep{Font-Ribera14} & BAO+RSD & 0.15 - 1.85 & $\\checkmark$ & $\\checkmark$ & $\\checkmark$ & $3\\times18$\\\\\n\n\\end{tabular}\n\\label{tab:data}\n\\end{table*}\n\n\\subsection{Cosmic Chronometers}\n\nCosmic Chronometers (CCs) are tracers of the evolution of the differential age of the Universe as a function of redshift. Since $H(z) \\equiv \\Dot{a}\/a = -(dt\/dz)\/(1+z)$ a measurement of dt\/dz directly yields the expansion rate \\citep{0106145}.  By measuring the age difference between two ensembles of passively evolving galaxies at different redshifts, one can determine the derivative of redshift with respect to cosmic time, $dz\/dt$. Massive, early, passively-evolving galaxies have been found to be very good tracers in this sense \\citep{Cimatti06, Thomas11, Moresco15, Moresco18, Moresco20} and have been used extensively over the past two decades to measure $H(z)$ up to $z \\approx 2$. In this work we make use the $H(z)$ measurements from CCs summarized in Table 1 of \\citet{2011.11645}.\n\n\n\\subsection{Supernovae}\n\\label{subsec: Supernovae}\nType Ia supernovae (SNe Ia) \u2014 titanic explosions of white dwarfs in multi-star systems  \\citep{Hoyle60, Colgate69}, are highly prized observations in cosmology due to their capacity to act as standard candles \\citep{9907052, 0012376}.\n\nHowever, SNe Ia by themselves can only inform their relative distance to one another and need to be calibrated with nearby SNe Ia of known redshift and luminosity distance $D_L$ to obtain absolute distances. Thus, SNe Ia inform the relationship\n\\begin{equation} \\label{eq: light_curve}\n    \\mu(z) = 5 \\log_{10} D_L(z) + 25 + M \\,, \n\\end{equation}\nknown as the luminosity distance modulus where $M$ is the calibrator known as the absolute magnitude of the SNe Ia. Therefore, once $M$ has been determined SNe Ia luminosity distance measurements can be used to inform the Hubble rate.\n\nIn this work, we fit the compressed data vector of the Pantheon sample, known as \"DS17\", composed of 40 measurements of the distance modulus (See Eqn. \\ref{eq: light_curve}) in the range $0.15\\leq z\\leq1.615$ \\citep{1710.00845}. The original Pantheon sample is composed of the optical light curves and redshifts for 365 spectroscopically confirmed Type Ia supernovae (SNe Ia) discovered by the Pan-STARRS1 (PS1) Medium Deep Survey combined with the subset of 279 PS1 SN Ia ($0.03 < z < 0.68$) with useful distance estimates of SN Ia from SDSS, SNLS, various low-z and HST samples to form the largest combined sample of SN Ia consisting of a total of 1048 SN Ia ranging from $0.01 < z < 2.3$ \\citep{1710.00845}. \n\nIn the light of recent works in the literature questioning the accuracy of the absolute calibration of important sectors of Pantheon sample \\citep{H0_ofer}, we marginalize over the absolute magnitude of the supernovae as opposed to fixing its value. Due to the degeneracy between $M$ and $H_0$, this is equivalent to fitting the expansion rate, $E(z)= H(z)\/H_0$, as opposed to the Hubble rate, $H(z)$.\n\n\\subsection{Baryon Acoustic Oscillations}\nBaryon acoustic oscillations (BAOs) enhance matter overdensities  at a characteristic physical separation scale which corresponds to the size of the sound horizon at the end of the drag epoch, $r_{\\rm{s}}(z_{\\rm d})$ \\citep{Peebles70, 0110414}. The sound horizon is defined as the distance a pressure wave can travel from its time of emission in the very early Universe up to a given redshift. This can be expressed as \n\\begin{equation} \\label{eq:rsound}\n    r_{\\rm{s}}(z)= \\int^{\\infty}_z \\frac{c_{\\rm{s}}  \\, {\\rm d}z'}{H(z')} \\;,\n\\end{equation}\nwhere $c_{\\rm s}$ denotes the speed of sound, and where $H(z)$ is the expansion rate at redshift $z$. The end of the drag epoch is defined as the time when photon pressure can no longer prevent gravitational instability in baryons around $z \\sim 1020$ \\citep{0803.0547}. \n\nThe BAO feature can be measured in the directions parallel and perpendicular to the line of sight. Perpendicular to the line of sight, the BAO feature informs the trigonometric relationship\n\\begin{equation}\\label{eq:acoustic}\n    \\theta = \\frac{r_{\\rm{s}}(z_{\\rm d})}{D_M(z)} \\;,\n\\end{equation}\nwhere $\\theta$ is the angle under which the scale of the sound horizon is observed.  Parallel to the line of sight, the BAO feature informs the relationship $ \\Delta z = H(z) r_{\\rm{s}}(z_{\\rm{d}})$ which can be used to constrain the expansion history of the Universe directly.\n\nIn this work, we make use of the twelfth data release of the galaxy clustering data set of the Baryon Oscillation Spectroscopic Survey (BOSS DR12) which forms part of the Sloan Digital Sky Survey (SDSS) III. BOSS DR12 comprises 1.2 million galaxies over an area of 9329 deg$^2$ and volume of 18.7 Gpc$^3$, divided into three partially overlapping redshift slices centred at effective redshifts 0.38, 0.51, and 0.61.\n\nWe fit the Alcock-Paczynski (AP) parameters  $\\alpha_\\parallel$ and $\\alpha_\\perp$ as reported by the BOSS DR12 data products\n \\begin{equation} \\label{eq:alphas_B}\n    \\frac{[H(z)]_{\\rm{fid}}}{\\alpha_\\parallel} = \\frac{H(z) [r_{\\rm{s}}(z_{\\rm{d}})]_{\\rm{fid}} }{ r_{\\rm{s}}(z_{\\rm{d}})}, \\quad\n    \\alpha_\\perp [\\chi(z)]_{\\rm{fid}} = \\frac{\\chi \/ r_{\\rm{s}}(z_{\\rm{d}})}{[r_{\\rm{s}}(z_{\\rm{d}})]_{\\rm{fid}}} ,\n\\end{equation}\nfrom the reconstruction of the BAO feature at the three different redshift bins where $[r_{\\rm{s}}(z_{\\rm{d}})]_{\\rm{fid}} = 147.78$ Mpc is the scale of the sound horizon at drag epoch as given by the fiducial cosmology used for the reconstruction. $[H(z)]_{\\rm{fid}}$ and $[\\chi(z)]_{\\rm{fid}}$ are the corresponding Hubble parameter and comoving radial distance for the fiducial cosmology, respectively. \n\nIn addition to this, we employ the  anisotropic clustering of quasars in the sixteenth data release of the extended Baryon Oscillation Spectroscopic Survey (eBOSS DR16 \\citealp{1703.00052}), which forms part of the Sloan Digital Sky Survey (SDSS) IV  \\citep{1508.04473}. The eBOSS DR16 catalog contains 343,708 quasars  between $0.8 < z < 2.2$, from which BAO and RSD meausurements are obtained at an effective redshift of $z_{\\rm eff} = 1.48$ \\citep{2007.08998}.  We use the results from the configuration space analysis performed by measuring the two-point correlation function and decomposing it using the Legendre polynomials.  Similarly to BOSS DR12, the BAO signal is measured both parallel and perpendicular to the line of sight. This allows for the measurement of the geometrical relationships $D_H(z_{\\rm eff})\/[r_{\\rm d}]_{\\rm fid}$ and $D_M(z_{\\rm eff})\/[r_{\\rm d}]_{\\rm fid}$ respectively, where $D_H(z) \\equiv c\/H(z)$ and $[r_{\\rm d}]_{\\rm fid} = 147.3$.\n\nFinally, we make use of the \\textit{Planck} 2018 measurement of the BAO angular scale $\\theta_* = \\frac{D_{\\rm{A}}(z_*)}{r_{\\rm{s}}(z_*)}$, where $z_* \\sim 1100$ is the redshift of the last scattering surface. We use the \\textit{Planck} measurement from the temperature and polarization maps denoted as TTTEEE + lowE.\n\n\\subsection{Redshift Space Distortions}\nRedshift space distortions (RSDs) are modifications to the observed redshift of a given object caused by its radial peculiar velocity \\citep{kaiser87}. RSDs are caused by deviations\nfrom the Hubble flow that are gravitationally induced by inhomogeneities in the gravitational potential of the surrounding matter distribution. On large, linear scales RSDs are dominated by the infall towards overdense structures, known as the Kaiser effect \\citep{Hamilton98}. Clustering two-point statistics as a function of transverse and line-of-sight separation are sensitive to the quantity $f \\! \\sigma_8$ via RSDs \\citep{1607.03150}.\nAlternatively, SNe Ia themselves can be used as a probe of the local velocity field \\citep{2105.05185, 9908237, 1605.01765}, which can also be used to measure this parameter.\n\nWe use the three measurements of  $f\\sigma_{\\rm{8}}(z)$ from RSD from the BOSS DR12 data, obtained using the anisotropic clustering of the pre-reconstruction density field \\citep{1607.03155}. We also include the value of $f \\! \\sigma_8 (z_{\\rm eff})$ measured from the BOSS DR16 quasar sample. The BAO and RSD measurements of both datasets are extracted from the same set of observations. As such, they are not statistically independent, and we account for their full covariance matrix in our analysis \\citep{1607.03155,2007.08998}. In addition to these, we use the $f \\! \\sigma_8$ measurements reported by the WiggleZ Dark Energy Survey at the Australian Astronomical Observatory \\citep{Drinkwater10} at redshifts 0.44, 0.60 and 0.73.\n\nFinally, we also use the value of $f \\! \\sigma_8 (z=0)$ derived from the measured peculiar velocities of the Democratic Samples of Supernovae (DSS), the largest  catalog used to study bulk flows in the nearby Universe, compiled of 775 low-redshift Type Ia and II supernovae (SNe Ia $\\&$ II). \\dam{You need to include a citation here!}\n\n\\subsection{Synthetic Stage-IV Data}\n\\label{sec: forecasts}\nIn addition to the previously discussed observables and data sets, we also produce forecasts for future experiments, focusing on the Dark Enegy Spectroscopic Instrument (DESI). DESI is a galaxy and quasar redshift survey currently taking data from the Mayall 4 meter telescope at Kitt Peak National Observatory. The baseline area is 14000 sq. deg. We use the forecast errors for the measurements of the Hubble rate, $H(z)$; the angular diameter distance, $D_A(z)$; and the growth rate measurement $f \\! \\sigma_8$ reported by \\citet{Font-Ribera14} to create a set synthetic measurements for $H(z)$, $D_A(z)$, and $f \\! \\sigma_8$ at an array of 18 redshifts from 0.15 to 1.85. The synthetic measurements were generated using the best-fit \\textit{Planck} 2018 cosmology, including measurement noise following the statistical uncertainties reported by \\citet{Font-Ribera14}.\n\n\\section{Results} \\label{Sect: Results}\n\n\\begin{table} \n\\caption{One dimensional constraints on the hyperparameter $\\eta$ (first column) and the mean variance of $\\delta H(z)$ (second column) for different analyses.}\n\\begin{tabular}{|p{3.2cm}|p{2cm}|p{2cm}|}\n \\hline\n  Analysis & $\\eta$ & $\\overline{\\sigma}(\\delta H(s))$ \\\\\n    \\hline\n    All data ($\\Lambda$CDM) & $0 \\pm 0$ & $0.011 \\pm 0.003$ \\\\\n    DESI+CMB ($\\Lambda$CDM) & $0 \\pm 0$ & $0.002 \\pm 0.001$ \\\\\n    \\hline\n    All data & $0.113 \\pm 0.075$ & $0.094 \\pm 0.004$ \\\\\n    All data (Fixed HP) & $0.2$ & $0.2$ \\\\\n    All data (Two-Steps) & $0.044 \\pm 0.039$ & $0.025 \\pm 0.01$ \\\\\n    No CMB & $0.142 \\pm 0.096$ & $0.125 \\pm 0.016$ \\\\\n    No DSS & $0.111 \\pm 0.074$ & $0.091 \\pm 0.005$ \\\\\n    Growth data & $0.147 \\pm 0.114$ & $0.183 \\pm 0.003$ \\\\\n    Geometry data & $0.122 \\pm 0.077$ & $0.098 \\pm 0.005$ \\\\\n    \\hline\n    DESI $+$ CMB & $0.085 \\pm 0.074$ & $0.08 \\pm 0.004$ \\\\\n\\end{tabular}\n\\label{tab:variance}\n\\end{table}\n\nIn this section we present the results of our analysis, including both the constraints on expansion history and cosmological parameters from current data, and forecasts for ongoing and future surveys. \n\n\\subsection{Validity of the methodology}\n\nBefore discussing the results of our analysis, we must first verify its reliability. Our methodology produces two types of results: constraints on the GP used to model the expansion history, and on the cosmological parameters (mainly $\\Omega_M$) involved in solving the Jeans equation (Eqn. \\ref{eq:growth}). We thus report results on both aspects. In App. \\ref{app: Alternative analyses} we present the results of the alternative analyses described at the end of Sect. \\ref{Sect: Methodology} and show that, despite the different treatments of the GP, they produce statistically compatible constraints for the cosmological functions as well as for the cosmological parameters, and similar uncertainties on both.  In App. \\ref{app:validation}, we considered five different cosmologies, each progressively more discrepant with our fiducial cosmology (\\textit{Planck} 2018), and generated mock data based on each them. We then showed that our methodology recovers the input cosmologies within statistical uncertainty. In addition to this, in App. \\ref{app: Using HMC}, we discuss the specific \\textsc{NUTS} set up used to produce our results. These settings allow us to achieve a Gelman-Rubin convergence test~\\citep{GelmanRubin} value of $R -1 < 0.01$ for all the combinations of data considered.\n\n\\subsection{Current constraints}\n\\subsubsection{Cosmological functions}\n\nBefore presenting the results for the cosmological functions of interest; i.e. the expansion history $H(s)$ and linear growth rate $f \\! \\sigma_8$, we must first discuss the results on the core of our analysis, the GP on $\\delta H(s)$. We found that our analysis of current data produces $\\delta H(s)$ constraints compatible with $0$ at all redshifts at less than $1\\sigma$ deviation. This means that the mean of the GP, $H_m(s) = A_0 \\overline{H}(s)$, is capable of capturing the long range correlations of the data allowing the GP to fit local features. The recovered bounds on $\\delta H$ are shown in Fig. \\ref{fig:dH_geo_vs_gro}.\n\nMoreover, we quantified how well different combinations of data can constrain $\\delta H(s)$. In other words, we measured how strong the agreement of $\\delta H(s)$ with zero is, and how it is affected by the data considered and the analysis choices. However, the fact that $\\delta H(s)$ has a multivariate distribution means that there is not a unique figure of merit for how well data constrains it. In this work we focused in two numbers. On the one hand, we looked at the hyperparameter $\\eta$ that constrains the prior distribution of values that $\\delta H(s)$ can take. On the other hand, we computed the mean variance of the $\\delta H(s)$ samples across redshift. In other words, for each parameter of $\\delta H(s)$ (i.e. $\\{\\delta H(s_1),\\,\\delta H(s_2),\\,...,\\,\\delta H(s_{200})\\}$), we computed the variance of the \\textsc{HMC} samples. Then, we took mean value of those variances, which is equivalent to averaging over redshift. We refer to this statistic as $\\overline{\\sigma}(\\delta H(s))$. The motivation behind $\\overline{\\sigma}(\\delta H(s))$ lies in the fact that it directly translates to average fractional constraints on $H(z)$ that can be readily interpreted. \n\nA summary of the impact of the different data sets and analysis choices on the distribution of $\\eta$ and $\\overline{\\sigma}(\\delta H(s))$ can be found in Tab. \\ref{tab:variance}. For the combination of data sets employed in our fiducial analysis, we find $\\eta = 0.113 \\pm 0.075$ and an $\\overline{\\sigma}(\\delta H(s)) = 0.094 \\pm 0.004$ corresponding to an average $9.4 \\pm 0.4\\%$ constraint on $H(z)$ across redshift. Removing the CMB data point significantly worsened both constraints finding $\\eta = 0.142 \\pm 0.096$ and an average $12.5 \\pm 1.6\\%$ constraint on $H(z)$ across redshift. In order to better understand this effect we can look at the first panel in Fig. \\ref{fig:dH_geo_vs_gro}. In this figure, we can observe how removing the CMB data from the analysis significantly widens the constraints of the GP, specially beyond $z>2.5$ (see Table \\ref{tab:variance}). This is to be expected as the CMB data is the only point we have above $z=2.5$. Thus, the constraints from this integrated effect are expected to become dominant in the redshift range between $2.5 < z < 1100$. However, it is important to note that the CMB is not the only contributor to the $\\delta H(s)$ constraints over this redshift range since $f \\! \\sigma_8$ data also constraints $\\delta H(s)$ over its whole domain through its role in solving the Jeans equation. The associated constraint is however very weak.\n\nFor the purpose of studying the effect of different data types, we split the data points within the data sets employed in the fiducial analysis in two groups: ``geometry'' -- exclusively containing measurements of the expansion history -- and ``growth'' -- solely containing $f \\! \\sigma_8$ measurements. \n\n\\begin{figure} \n\\centering\n    \\includegraphics[width=\\linewidth]{{\"dH_gp_geo_vs_gro\"}.png} \n    \\caption{$1\\sigma$-constraints on $\\delta H(z)$ broken down by type of data considered. Solid lines represent the mean of the GPs at each redshift. In red we display the constraints resulting from the analysis of all present data, in blue the effect of removing the CMB data set, in black the effect of fixing the $\\Omega_m$ and $\\sigma_8$ to their best-fit (BF) value; in magenta, the constraints resulting of only considering growth data; and in green, the constraints resulting of only considering geometry data.}\n    \\label{fig:dH_geo_vs_gro}\n\\end{figure}\n\nAs we can see in the second panel of Fig. \\ref{fig:dH_geo_vs_gro}, growth data only is much weaker  at constraining $\\delta H(z)$. From Table \\ref{tab:variance} we see that constraints from growth data alone on $\\overline{\\sigma}(\\delta H(s))$ are approximately twice as wide as those resulting from analysing the entire data set. These constraints are consistent with the prior on the hyperparameter $\\eta$. On the other hand, the constraining power of the geometry data is only slightly weaker than that of the entire data set. Hence the $\\delta H(z)$ constraints are mostly dominated by the geometry data sets as one would expect. Nonetheless, the addition of growth data increases constraining power. This is shown explicitly in the last panel of Fig. \\ref{fig:dH_geo_vs_gro}, which shows the results of using geometry data alone compared to those using the full data set. This recovers the expected behavior: more data increases the constraining power and the contours shrinks.\n\n\\begin{figure} \n\\centering\n    \\includegraphics[width=\\linewidth]{{\"data_cosmo_functions\"}.pdf} \n    \\caption{$1\\sigma$-constraints for the cosmological functions $H(z)$ and $f \\! \\sigma_{8}$ (top and bottom panel respectively) broken down by combination of data set. Solid lines represent the mean of the GPs at each redshift. The dashed black lines show the prediction for each cosmological function using our Planck 2018 fiducial cosmology (see Table~\\ref{tab:fiducial}). In red we show the constraints resulting of the analysis of all present data; in green, the impact of removing the CMB data set; and in black, the impact of removing the DSS data set.}\n    \\label{fig:comp_data}\n\\end{figure} \n\nWe now shift the focus of our discussion to the constraints we derive from $\\delta H(s)$ for the expansion history itself, $H(z)$, and the linear growth of matter anisotropies, $f\\!\\sigma_8$. Comparing the constraints for both cosmological functions from our analysis of current data with the \\textit{Planck} 2018 predictions, we find an overall good agreement, finding both functions to contain the \\textit{Planck} 2018 predictions within their $2\\sigma$ confidence contours. This can be seen in  Fig. \\ref{fig:comp_data}. Nonetheless, we observed a greater than 1$\\sigma$ preference for a lower $f\\!\\sigma_8$ between $0<z<0.75$, mostly driven by the DSS data point. However, the constraining power of current $f\\!\\sigma_8$ data is too weak to make a case for the presence of new physics. \n\n\\begin{figure*} \n\\hspace*{-0.7cm} \n\\centering\n    \\includegraphics[width=\\linewidth]{{\"1D_comp\"}.pdf} \n    \\caption{Comparison between the different $1 \\sigma$ constraints for the cosmological parameters $S_8$,  $\\sigma_{8}$ and $\\Omega_m$ (first, second and third panel respectively). In each panel the \\textit{Planck} 2018 constraint is displayed in the form of a blue band for comparison.}\n    \\label{fig:1D_comp}\n\\end{figure*} \n\n\\subsubsection{Cosmological parameters}\nWe now focus on the constraints on the specific cosmological parameters $\\Omega_M$ and $\\sigma_8$. We will also show constraints on the derived parameter $S_8 = \\sigma_8 \\sqrt{ \\Omega_m \/ 0.3}$ and on the local value of the expansion rate, given in our case by $H_{0}\\equiv A_0 \\, \\overline{H}_0 \\, (1+ \\delta H(s=0))$. A summary of the constraints obtained by the different analyses considered in this work can be found in Table \\ref{tab:constraints} with a graphical illustration in Fig \\ref{fig:1D_comp}. Full 2D contours for the respective sets of constraints, including hyperparameters, can be found in Appendix \\ref{app:full_results}. We primarily focus our discussion on the cosmological parameter $\\Omega_m$, the only remaining degree of freedom in Eqn. \\ref{eq:growth}. \n\n\\begin{table*} \n\\caption{Constraints on the cosmological parameters $\\Omega_m$, $\\sigma_8$, $S_8$,  and $H_0$ (first to fourth columns respectively) for each of the different analyses (rows), as well as the \\textit{Planck} 2018 constraints and $\\Lambda$CDM analyses of current and DESI+CMB for reference (first to third row respectively).}\n\\begin{tabular}{|p{3cm}|p{1.75cm}|p{1.75cm}|p{1.75cm}|p{3cm}| }\n \\hline\n  Analysis & $\\Omega_m$ & $\\sigma_{8}$ & $S_8$ & $H_0 = \\Bar{H_{0}} A_0 (1+\\delta H(0))$ \\\\ \n    \\hline\n    \n    \\textit{Planck} 2018 &   $0.317 \\pm 0.008$ &  $0.812 \\pm 0.007$  & $0.834 \\pm 0.016$ &  $67.27 \\pm 0.006 $\\\\\n    All data ($\\Lambda$CDM) & $0.283 \\pm 0.007$ & $0.8 \\pm 0.029$ & $0.777 \\pm 0.028$ & $68.601 \\pm 0.775$ \\\\\n    DESI+CMB ($\\Lambda$CDM) & $0.316 \\pm 0.006$ & $0.812 \\pm 0.003$ & $0.834 \\pm 0.008$ & $66.992 \\pm 0.311$ \\\\\n    \\hline\n    All data & $0.224 \\pm 0.066$ & $0.946 \\pm 0.158$ & $0.788 \\pm 0.029$ & $67.715 \\pm 0.93$ \\\\\n    All data (Fixed HP) & $0.228 \\pm 0.067$ & $0.932 \\pm 0.155$ & $0.785 \\pm 0.03$ & $68.113 \\pm 1.061$ \\\\\n    All data (Two-Steps) & $0.226 \\pm 0.066$ & $0.936 \\pm 0.151$ & $0.787 \\pm 0.029$ & $68.144 \\pm 0.718$ \\\\\n    No CMB & $0.227 \\pm 0.068$ & $0.936 \\pm 0.156$ & $0.786 \\pm 0.03$ & $67.94 \\pm 1.034$ \\\\\n    No DSS & $0.229 \\pm 0.106$ & $0.974 \\pm 0.218$ & $0.791 \\pm 0.047$ & $67.766 \\pm 0.96$ \\\\\n    DESI $+$ CMB & $0.293 \\pm 0.013$ & $0.839 \\pm 0.014$ & $0.828 \\pm 0.008$ & $66.788 \\pm 0.371$ \\\\\n\\end{tabular}\n\\label{tab:constraints}\n\\end{table*}\n\nTo establish a benchmark against which to compare our method, we start by examining the constraints obtained assuming a $\\Lambda$CDM model in which we vary $H_0$, $\\Omega_m$ and $\\sigma_8$. This allows us to quantify the impact of performing a model-independent analysis using GPs on the final constraining power. Looking at Fig. \\ref{fig:1D_comp} and Table \\ref{tab:constraints} we can observe that, assuming $\\Lambda$CDM,  $\\Omega_m = 0.283 \\pm 0.007$. In other words, for the setup used in this work, we can constrain $\\Omega_m $ to around $2\\%$ precision if we undertake a model-dependent analysis with  $\\Lambda$CDM. In this case, $\\Omega_m$ receives information from both background and perturbations.\n\nIn turn, our fiducial model-independent analysis yields $\\Omega_m = 0.224 \\pm 0.066$, inflating the uncertainty by a factor $\\sim9$. Comparing this result with the \\textit{Planck}  2018 $\\Omega_m = 0.317 \\pm  0.008$, our $\\Omega_m$ constraint is lower but statistically compatible with the \\textit{Planck} 2018 constraint and our $\\Lambda$CDM analysis both at $1.5 \\sigma$.\n\nLooking at the relevance of the different individual data sets on the constraints, we observe an excellent agreement between all the different combinations considered (see Fig. \\ref{fig:1D_comp} and Tab. \\ref{tab:constraints}).  Removing the DSS data, one of the most precise $f \\! \\sigma_8$ measurements, significantly worsens the $\\Omega_m$ constraints by nearly $60 \\%$. On the other hand, removing the CMB data set resulted in nearly identical constraints for the cosmological parameters. This is due to the fact that, in the presence of other geometry data to inform the expansion history, $\\Omega_m$ constraints become dominated by growth data through the relationship between both. Thus, it is fair to ask what the impact of completely removing any type of geometry data from the analysis is. Analytically, we can see from Eqn. \\ref{eq:jeans} that in the presence of measurements of the linear growth rate and an arbitrary value of $\\Omega_m$ one can always find a expansion history that solves the differential equation. This degeneracy between the expansion history and $\\Omega_m$ in Eqn. \\ref{eq:jeans}, prevents our methodology from obtaining any meaningful constraints on $\\Omega_m$ in the absence of Geometry data that is not completely dominated by our choice of GP mean and hyperparameter priors.\n\nLooking at the parameters $\\sigma_8$ and $S_8$ in more detail, we found compatible constraints with the \\textit{Planck} 2018 cosmology up to 1$\\sigma$. However, it is worth mentioning that our results show a mild tendency towards higher $\\sigma_8$ values which, combined with the tendency towards lower  $\\Omega_m$ values, results in lower $S_8$ values. This is consistent with the underprediction of $f \\! \\sigma_8$ between $0<z<0.75$ we discussed in the previous section. A lower value of $S_8$ would also be consistent with the most recent measurements by large scale structure experiments \\citep{K1K, DESY3, Garcia-Garcia21, 2021arXiv211109898W, 2021arXiv210503421K}, the origin of which could lie on a lower value of $\\Omega_m$ \\citep{2021MNRAS.501.1481H}.\n\n\\subsection{Forecasts}\nOur model-independent analysis leads to far weaker constraints than assuming the $\\Lambda$CDM model. Being data-driven, the performance of the method used here may improve significantly with the advent of next-generation surveys with significantly tighter uncertainties. To quantify this, we repeated our fiducial analysis pipeline on mock data generated based on the forecasted errors for the DESI mission \\citep{Font-Ribera14} in combination with the \\textit{Planck} measurement of $\\theta_*$. The results can be found in Fig. \\ref{fig:dH_forecast}. In this figure we can observe a $10\\%$ improvement between the expected $\\delta H(z)$ constraints from DESI with respect to those of current data. The corresponding $\\eta$ and $\\overline{\\sigma}(\\delta H(s))$ constraints (shown in Tab. \\ref{tab:variance}) improve by approximately $15\\%$: $\\eta = 0.085 \\pm 0.074$ and $\\overline{\\sigma}(\\delta H(s)) = 0.080 \\pm 0.004$. Looking at the cosmological parameters,  DESI \\citep{Font-Ribera14} in combination with the CMB data set results in nearly 5 times tighter $\\Omega_m$ constraints and 10 times better constraints on $\\sigma_8$.\n\n\\begin{figure} \n\\centering\n    \\includegraphics[width=\\linewidth]{{\"dH_gp_forecast\"}.png} \n    \\caption{Forecast $1\\sigma$-constraints on $\\delta H(z)$ from DESI (blue) and using current data (red). Solid lines represent the mean of the GPs at each redshift.}\n    \\label{fig:dH_forecast}\n\\end{figure}\n\nWhile these constraints are still significantly worse than the \\textit{Planck} 2018 results, they are comparable if not better than using the $\\Lambda$CDM model to analyze the currently available data used in this work. For example, while GP constraints on $\\Omega_m$ from DESI are two times wider than those currently found assuming a $\\Lambda$CDM model, the constraints on $\\sigma_8$ and $S_8$ are 2.3 and 3.6 times tighter respectively. \n\nIf we, instead, compare how a model-independent analysis of DESI data using this methodology will pitch against a $\\Lambda$CDM analysis, we find that the gap between the model-independent and the model-dependent constraints shrinks. This can be see in the fact that, with current data, the $\\Lambda$CDM constraint on $\\Omega_m$ is $9.2$ times tighter than the one obtained with our methodology. However, in our DESI forecast, it is only twice as good. Similarly, the improvement for the parameter $\\sigma_8$ reduces from five times tighter constraints to about four times. Our understanding is that, as better data allow us to better reconstruct $H(z)$, the difference between the reconstructed and model-dependent $H(z)$ shrinks, as long as the assumed model fits the data well. As a consequence, the constraints on the other parameters ($\\Omega_m$, $\\sigma_8$ and $S_8$) become more similar. Thus, we expect that in the future, as the quality of the data keeps improving, cosmological constraints from model-independent methodologies, such the one proposed in this work, will rival model-dependent ones.\n\n\\section{Conclusions} \n\\label{Sect: Conclusions}\n In this work we have developed a method to obtain  constraints on $H(z)$ and $\\Omega_m$ purely based on the relationship between the expansion history and the linear growth rate. In order to do so, we employ a Gaussian process to model the evolution of the expansion history from present time to recombination.\n\nFrom this expansion history, we have derived predictions for the comoving distance and the linear growth rate $f \\! \\sigma_8$ making use of the physical relationships between the three quantities. Constraints for $\\Lambda$CDM parameters were obtained by simultaneously fitting a suite of the latest measurements of these three cosmological functions. The data combination used for our fiducial analysis consisted of cosmic chronometers, the Pantheon supernova catalog, BAO and RSD data, peculiar velocity data from supernovae, and the position of the first acoustic peak in the CMB power spectrum. Moreover, we also obtained forecast constraints on these cosmological functions from the future DESI data.\n\nCurrent data can constrain the $H(z)$ Gaussian process at an average $9.4\\%$ throughout all redshifts. These constraints are compatible up to $1 \\sigma$ with the best-fit \\textit{Planck} 2018 expansion history across $0 < z < 1100$. Our constraints on the expansion rate $f \\! \\sigma_8$ lie below the corresponding \\textit{Planck} prediction by less than 2-$\\sigma$ in the range $0 < z < 0.75$ (a result driven by the Democratic Supernova Sample data).\n\nTranslating the Gaussian process constraints into constraints of cosmological parameters, we find a model-independent measurement of $\\Omega_m = 0.224 \\pm 0.066$. This result is lower than, but statically compatible with, the Planck 2018 \\citep{Planck} cosmology. We also find $S_8 = 0.788 \\pm 0.029$, an intermediate value, statistically compatible with both the Planck 2018 cosmology \\citep{Planck} and recent local measurements from weak lensing and galaxy  clustering\\citep{DESY3, K1K, Garcia-Garcia21, 2021arXiv210503421K, 2021arXiv211109898W}.\n\nThe forecast analysis performed using the methodology of this work predicts that combining the DESI measurements with the CMB BAO data used in this work will yield $15\\%$ tighter constraints on $H(z)$ across redshift as well as five times tighter constraints on $\\Omega_m$. While this constraints are still significantly looser than the model-dependent cosmological constraints of \\textit{Planck} 2018, we show that a model-independent analysis of DESI plus CMB BAO data would be as powerful as a $\\Lambda$CDM analysis of current data. Indeed, we find that model-independent constraints using this methodology would achieve four times tighter $S_8$ constraints. Thus, in the future it will be possible to weigh in on the ongoing $S_8$ tension making use of model-independent methods. Moreover, we also show that, as the quality of the data increases as we go into the future, the gap between the constraining power of model-independent and model-dependent constraints will significantly shrink. \n\nFuture implementations of this methodology could explore several possible extensions. On the one hand, the types of data used in this work are greatly limited by lack of differentiable tools to obtain theoretical predictions for observables with greater constraining power. The development of tools such as differentiable Boltzmann codes, differentiable emulators of the non-linear matter power spectrum, or differentiable Limber integrators would allow us to fit the power spectrum data directly. These developments would enable model-agnostic analyses similar to that of \\citet{Garcia-Garcia21} but with a greatly reduced number of assumptions and a more reliable measure of uncertainty in their results. On the other hand, it would also be possible to explore the use of Gaussian processes to constrain general forms of modified gravity, generalizing works such as \\citet{2017PhRvD..96h3509R, 2019PhRvD..99b3512E, 2021PhRvD.103j3530P, 2021arXiv210712990R, 2021arXiv210712992P}, and study how these theories can be informed by the relationship between background and perturbations. Gaussian processes are an exceptional tool to constrain modified gravity since they don't require assuming a particular parametrisation of such deviations. A comprehensive list of different departures from $\\Lambda$CDM that could be explored with a similar methodology to the one presented in this work can be found in \\citet{Baker14}.\n\n\\section*{Acknowledgements}\n\\textit{Author contributions}: All authors contributed to the development and writing of\nthis paper. We acknowledge support from the Beecroft Trust. CGG and PGF acknowledge funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 693024). DA acknowledges support from the Science and Technology Facilities Council through an Ernest Rutherford Fellowship, grant reference ST\/P004474. JRZ is supported by an STFC doctoral studentship.\n\nThe analysis made use of the software tools\n\\textsc{SciPy} \\citep{scipy}, \\textsc{NumPy} \\citep{np}, \\textsc{Matplotlib} \\citep{np},\n\\textsc{CLASS} \\citep{Class}, GetDist \\citep{getDist}.\n\nBased on observations obtained with \\textit{Planck} (\\url{http:\/\/www.esa.int\/Planck}), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada.\n\n\nFunding for the Sloan Digital Sky Survey IV has been provided by the  Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and  resources from the Center for High Performance Computing  at the University of Utah. The SDSS website can be found at \\url{www.sdss.org}.\nSDSS-IV is managed by the  Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including  the Brazilian Participation Group,  the Carnegie Institution for Science,  Carnegie Mellon University, Center for Astrophysics | Harvard \\& Smithsonian, the Chilean Participation, the French Participation Group,  Instituto de Astrof\\'isica de Canarias, The Johns Hopkins  University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) \/ University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut f\\\"ur Astrophysik Potsdam (AIP),  Max-Planck-Institut f\\\"ur Astronomie (MPIA Heidelberg), Max-Planck-Institut f\\\"ur  Astrophysik (MPA Garching), Max-Planck-Institut f\\\"ur Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observat\\'ario Nacional \/ MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Aut\\'onoma de M\\'exico, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.\n\n\\section*{Data Availability}\nThe code developed for this work as well as the data sets used are available upon request. \n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nStrong spin-orbit coupled unconventional superconductors, whose superconducting (SC) state cannot be described by electron-phonon coupling, provide a platform for experimental and theoretical studies of emergent quantum behavior \\cite{Pfleiderer2009,Smidman2017}. Time-reversal and parity are key symmetries to characterize these materials, and striking states of matter often emerge when one (or both) of these symmetries are broken. For instance, odd-parity superconductors have been identified as a promising route for topological superconductivity, which hosts edge modes or vortices with non-abelian statistics required for topological quantum computing~\\cite{SatoAndo2017}. A chiral superconductor further breaks time-reversal symmetry and lowers the energy of the SC condensate by removing nodes from the gap function~\\cite{Kallin2016}. Odd-parity chiral superconductors are remarkably rare, but their experimental manifestation has been observed in superfluid $^3$He and actinide superconductor UPt$_3$~\\cite{Avers2020Broken}. \n\nUTe$_2$ is a newly-discovered candidate for odd-parity chiral superconductivity~\\cite{Ran2019Nearly,Aoki2022Unconventional}. Nuclear magnetic resonance (NMR) Knight shift measurements strongly suggest that UTe$_2$ is an odd-parity superconductor with a dominant $B_{3u}$ order parameter~\\cite{Nakamine2019Sup,Nakamine2021Ani,Fujibayashi2022Super}. Point nodes in the SC gap structure are supported by transport measurements~\\cite{Ran2019Nearly,Metz2019Point,Kittaka2020Ori}, Knight shift measurements~\\cite{Fujibayashi2022Super}, and non-local superfluid density measurements~\\cite{Bae2021Ano}. \nThe position of the point nodes, however, is still controversial. Thermal conductivity, microwave surface impedance, and specific heat measurements suggest point nodes in the $ab$-plane or along $a$~\\cite{Metz2019Point,Bae2021Ano,Kittaka2020Ori}, whereas magnetic penetration depth measurements argue for a multicomponent SC state with multiple point nodes near the $b$- and $c$-axes~\\cite{Ishihara2021Chiral}.  \n\nEvidence for chiral superconductivity was found in UTe$_2$ by scanning tunneling spectroscopy on the step edges of a ($0\\bar{1}1$)-plane~\\cite{Jiao2020Chiral} and by polar Kerr rotation measurements~\\cite{Hayes2021Multi}. Multiple SC phase transitions were also reported in UTe$_2$ even at ambient pressure by specific heat measurements~\\cite{Hayes2021Multi, Thomas2020}. Subsequently, it was found that the observed ``double peak\" in the specific heat can arise from sample inhomogeneity~\\cite{Thomas2021Spat}. In addition, a single phase transition is reported in higher quality samples with higher  SC critical temperature  $T_{\\rm c}$, higher residual resistivity ratios, lower residual resistivities, and quantum oscillations~\\cite{Rosa2021Single,Aoki2022Unconventional}. \n\nTo microscopically investigate the SC state of UTe$_2$, here we report the temperature dependence of the local superfluid response using scanning SQUID (superconducting quantum interference device) susceptometry on a cleaved (011)-plane of UTe$_2$. We also image the pinned vortices induced by field cooling. Our results show no evidence for multiple phase transitions in the temperature dependence of the superfluid density and imply an anisotropic nodal gap structure in UTe$_2$.\n\n\\begin{figure*}[htb]\n\\begin{center}\n\\includegraphics*[width=16cm]{.\/Fig1.jpg}\n\\caption{ Local susceptibility is microscopically homogeneous on (011) surface of UTe$_2$ sample\\#1. (a) Optical image of the scanned area, which includes the cleaved (011) surface with small bumps creating no signals in our scans and the edge. (b-e) Temperature dependence of the susceptometry scan indicates the homogeneous superfluid density on UTe$_2$. Stripes along the scan directions are the scanning noise. (f) The temperature dependence of the local susceptibility at the points from A to A' in Fig. 1(e) has no kink below $T_{\\rm c}$. The susceptibilities are shifted by 0.2 $\\Phi_0$\/A for clarity except for the data at A.}\n\\end{center}\n\\end{figure*}\n\nBulk single crystals of UTe$_2$ were grown by chemical vapor transport. Samples \\#1 and \\#2 used in this paper were obtained from the same batch as sample s2 in Ref.~\\cite{Rosa2021Single}. Heat capacity measurements confirmed a single SC transition at $T_{\\rm c} =1.68$~K with a width of 50 mK on a single crystal which was subsequently cleaved into two samples used in this study. We used a scanning SQUID susceptometer to obtain the local ac susceptibility on a cleaved (011)-plane of UTe$_2$ at temperatures from 80 mK to 2 K in a Bluefors LD dilution refrigerator. Our scanning SQUID susceptometer has two pickup loop and field coil pairs configured with a gradiometric structure~\\cite{Kirtleyrsi2016}. The inner radius of the pickup loop is 0.4 $\\mu$m and the inner radius of the field coil is 1.5 $\\mu$m. The scan height is $\\sim$500 nm. The pickup loop provides the local dc magnetic flux $\\Phi$ in units of the flux quantum $\\Phi_0=h\/2e$, where $h$ is the Planck constant and $e$ is the elementary charge. The pickup loop also detects the ac magnetic flux $\\Phi^{ac}$ in response to an ac magnetic field $He^{i\\omega t}$, which was produced by an ac current of $|I^{ac}| =$ 1 mA at a frequency $\\omega\/2\\pi\\sim$ 1 kHz through the field coil, using an SR830 Lock-in-Amplifier. Here we report the local ac susceptibility as $\\chi=\\Phi^{ac}\/|I^{ac}|$ in units of $\\Phi_0$\/A.\n\n\nTo obtain the homogeneity of the superfluid density and its local temperature dependence, we measured the local susceptibility near the edge of the sample [Figs. 1(a)-1(e)]. The susceptibility far from the edge has a homogeneous temperature dependence on the micron-scale [Fig. 1(f)]. We note that our results do not rule out possible inhomogeneity either on the nanoscale or on scales larger than the scan area (e.g  sub-millimeter). Our data also cannot rule out fluctuations in time. There is no kink in the temperature dependence of the local susceptibility below $T_{\\rm c}$, wherein the temperature step of the scans is 25 mK. The susceptibility is positive above $T_{\\rm c}$ due to paramagnetism. The local susceptibility was negative (diamagnetic) near the edge but positive far from the edge at 1.69 K [Figs. 1(c),1(f)]. \n\n\n\nWe defined the local $T_{\\rm c}$ as the temperature that satisfies the relation of $\\chi(T>T_{\\rm c})>$-0.1 $\\Phi_0\/$A. The local $T_{\\rm c}$ mapping clearly shows that the local $T_{\\rm c}$ is weakly enhanced at the edge but is homogeneous 30~$\\mu$m away from the edge into the sample [Fig. 2, sample\\#1; Fig. S2, sample\\#2]. We note that the reported local $T_{\\rm c}$ near the edge is a lower bound relative to the actual value because the penetration depth near $T_{\\rm c}$ is longer than the pickup loop's scale and the susceptibility loses some signal at the edge. If two phase transitions do exist, they must be closer to each other than 25 mK, or the second kink below $T_{\\rm c}$ is much smaller than our experimental noise.\n\n\\begin{figure}[!hb]\n\\begin{center}\n\\includegraphics*[width=7cm]{.\/Fig2.jpg}\n\\caption{ Small enhancement of local $T_{\\rm c}$ near the edge of sample\\#1. (a) The local $T_{\\rm c}$ mapping is obtained from the local susceptometry scans. (b) Cross section of the local $T_{\\rm c}$ from A to A' shows the local $T_{\\rm c}$ enhancement of 30 mK at the edge. The plotted area and the cross section from A to A' are the same with Fig. 1.}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure*}[htb]\n\\begin{center}\n\\includegraphics*[width=13cm]{.\/Fig3.jpg}\n\\caption{The vortex density is homogeneous over many-micron distances. The existence of vortices and antivortices in low-field scans may indicate a local magnetic source in the sample\\#1. (a,b) Local magnetometry scan after field cooling shows the vortices pinned parallel to the sample edge, as denoted by the dashed lines. (c) There are a few vortices and antivortices pinned far from the edge after near zero field cooling. (d) Magnetometry scan near zero field above $T_{\\rm c}$ shows a small magnetic dipole at the sample's edge, but no other indication of magnetism. The \"tail\" of the vortices and dipoles are due to the asymmetric shielding structure of the scanning SQUID~\\cite{Kirtleyrsi2016}.}\n\\end{center}\n\\end{figure*}\n\nNow we turn to the pinned vortex density, which reflects the impurity density on the crystal surface for small applied magnetic fields. The distance between vortices is on the order of microns. Our magnetometry scan imaged the pinned vortices induced by cooling in an applied uniform magnetic field from 2 K to 100 mK [Figs. 3(a),3(b), sample\\#1; Figs. S3(a),S3(b), sample\\#2]. The number of vortices corresponds to the applied field, but the vortices are preferentially pinned along lines in one direction, which is parallel to the sample edge. This linear pinning indicates the existence of a line anomaly, such as nanometer-scale step edges along crystal axes. Near zero magnetic field, there are still a few vortices and antivortices pinned far from the edge [Fig. 3(c), sample\\#1; Fig. S3(c), sample\\#2]. Notably, these vortices and antivortices do not disappear after zero field cooling with slower cooling rates, which is expected to cancel the uniform background field normal to the sample surface by the application of an external field. These data are inconsistent with the argument from polar Kerr effect measurements that there are no vortices in UTe$_2$ within the beam size area ($\\sim11~\\mu$m radius)~\\cite{Wei2022}. Further, our results indicate the existence of a local magnetic source that induces vortices and antivortices, in spite of the absence of long-range order or strong magnetic sources on the scan plane above $T_{\\rm c}$ \\cite{Ran2019Nearly,Miyake2019Meta,Hutanu2020Low}. \nSmall dipole fields are observed at the edge of the sample, which may stem from U impurities; however, these impurities cannot induce pinned vortices and antivortices as they are too far away [Fig. 3(d), sample\\#1]. Muon spin resonance and NMR measurements have detected the presence of strong and slow magnetic fluctuations in UTe$_2$ at low temperatures~\\cite{Tokunaga2022Slow,Sundar2022Ubi}. Therefore, a sensible scenario is that these fluctuations are pinned by defects and become locally static.\n\n\n\\begin{figure*}[htb]\n\\begin{center}\n\\includegraphics*[width=16cm]{.\/Fig4.jpg}\n\\caption{The temperature dependence of the  superfluid density = best matches an anisotropic, rather than isotropic, gap structure. (a-c) Temperature dependence of the normalized superfluid density $n_{(011)}$ at the fixed position in sample\\#1,\\#2 that are indicated by the blue dot in Figs. 1(b), S1(b), respectively. The thick lines are simulation curves best fit for 4 gap symmetries with (a) Isotropic FS, (b) Ellipsoidal FS, and (c) Cylindrical FS~\\cite{supple}. (d-i) Best fit models of gap symmetries (d,e), (f,g) and (h,i) for (a), (b) and (c), respectively. FS is plotted by yellow color. The distance between larger surfaces and FS represents the angular dependence of the SC gap $\\Omega$ in (a,b) the spherical coordinate and (c) the cylindrical coordinate. All surfaces are cut for clarity.}\n\\end{center}\n\\end{figure*}\n\n\nTo estimate the local superfluid density, we measure the local susceptibility at different temperatures with the pickup loop position fixed. The local superfluid density is obtained using the numerical expression of the susceptibility assuming a homogeneous penetration depth, $\\lambda$, as described below. Kirtley {\\it et al}. developed the expression for the susceptibility as a function of the distance between the susceptometer and the sample surface~\\cite{Kirtley2012}. In this model, wherein the sample surface is at $z=0$, we consider three regions.  Above the sample ($z>0$), the pickup loop and field coil are at $z$ in vacuum and $\\mu_1 = \\mu_0$, where $\\mu_0$ is the permeability in vacuum. In the sample ($-t\\leq z \\leq 0$), the London penetration depth is $\\lambda=\\sqrt{m\/4\\pi ne^2}$, and the permeability is $\\mu_2$. Below the sample ($z<-t$), there is a nonsuperconducting substrate with a permeability $\\mu_3$. The radius of the field coil and the pickup loop are $a$ and $b$, respectively. By solving Maxwell's equations and the London equation for the three regions, the SQUID height dependence of the susceptibility $\\chi(z)$ is expressed as \n\\begin{widetext}\n\\begin{equation}\\label{phi-z}\n    \\chi(z)\/\\phi_s = \\int_0^\\infty{dx e^{-2x\\bar{z}}xJ_1(x)}\\left[\\frac{-(\\bar{q}+\\bar{\\mu}_2x)(\\bar{\\mu}_3\\bar{q}-\\bar{\\mu}_2x)+e^{2\\bar{q}\\bar{t}}(\\bar{q}-\\bar{\\mu}_2x)(\\bar{\\mu}_3q+\\bar{\\mu}_2x)}{-(\\bar{q}-\\bar{\\mu}_2x)(\\bar{\\mu}_3\\bar{q}-\\bar{\\mu}_2x)+e^{2\\bar{q}\\bar{t}}(\\bar{q}+\\bar{\\mu}_2x)(\\bar{\\mu}_3q+\\bar{\\mu}_2x)}\\right],\n\\end{equation}\n\\end{widetext}\nwhere $\\phi_s = A\\mu_0\/2\\Phi_0a$ is the self inductance between the field coil and the pickup loop, $A$ is the effective area of the pickup loop, $\\bar{z} = z\/a$, $J_1$ is the Bessel function of first order, $\\bar{t} = t\/a$, $\\bar{q} = \\sqrt{x^2 + \\bar{\\lambda}^{-2}}$, and $\\bar{\\lambda} = \\lambda\/a$. For the bulk sample on a copper substrate ($\\bar{t}>>1, \\mu_3 = 1$), the observed susceptibility only depends on $\\lambda$, $\\mu_2$, and the SQUID structure. \n\nThe penetration depth $\\lambda(T)$ was calculated using Eq.~\\eqref{phi-z} and the observed susceptibility [Fig. 4(a)]. The normalized superfluid density $n_{s} = \\lambda^2(0)\/\\lambda^2(T)$ was calculated from the obtained penetration depth's temperature dependence, where $\\lambda(0) = 1620\\pm150$~nm [sample\\#1], $1730\\pm300$~nm [sample\\#2] [Fig.~4(b)]. Here the error for $\\lambda$ and $n_s$ is roughly calculated from the pickup loop height uncertainty. We note that sample\\#2 had a dead layer of 700~nm on the surface, which we estimated by assuming that sample\\#2 has a similar penetration depth at zero temperature with sample\\#1, because sample\\#2 was accidentally exposed in air about one extra hour. The locally obtained superfluid density $n_s$ saturates below $T\/T_{\\rm c}=0.1$.\n\n\nWe examine the SC gap structure through the temperature dependence of the superfluid density. The superfluid density $n_{i}$ is sensitive to low-energy excitations along the $i$ axis, which is perpendicular to the applied field. In our case, $n_{i}$ is sensitive to excitations within the plane normal to [011], and the extracted $n_{(011)}$ is the average of $n_a$ and $n_{\\perp[011],a}$)~\\cite{Chandrasekhar1993}. The SC gap function of UTe$_2$, $\\Delta$, is most likely odd-parity within the orthorhombic $D_{2h}$ point group. In the presence of strong spin-orbit coupling, $\\Delta(T,\\Vec{k})=\\Psi(T)\\Omega(\\Vec{k})$, and the angle dependence of the gap function is expressed as $\\Omega(\\Vec{k})\\propto\\sqrt{\\Vec{d}\\cdot\\Vec{d}^*\\pm|\\Vec{d}\\times\\Vec{d}^*|}$. In this case, the possible irreducible representations are $A_{1u}$ [full gap, $\\Vec{d}=(c_1k_x,c_2k_y,c_3k_z)$)], $B_{1u}$ [point nodes along $c$, $\\Vec{d}=(c_1k_y,c_2k_x,c_3k_xk_yk_z)$], $B_{2u}$ [point nodes along $b$, $\\Vec{d}=(c_1k_z,c_2k_xk_yk_z,c_3k_x)$], and $B_{3u}$ [point nodes along $a$, $\\Vec{d}=(c_1k_xk_yk_z,c_2k_z,c_3k_y)$]~\\cite{Annett1990}. We note that coefficients $c_1, c_2$, and $c_3$ may differ by orders of magnitude~\\cite{IshizukaPRB2021}. \n\nFor the sake of completeness, here we assume three cases of Fermi surface structure to calculate the temperature-dependent superfluid density with fit parameters $c_1, c_2$, and $c_3$~\\cite{supple,Kogan2021}: (I) isotropic Fermi Surface (FS) based on the isotropic heavy 3D Fermi surface observed by angle-resolved photoemission spectroscopy (ARPES) measurements ~\\cite{FujimoriJPSJ2019,MiaoPRL2020}; (II) ellipsoidal FS based on the  upper critical field~\\cite{AokiJPSJ2019}; and (III) cylindrical FS. Case (III) is based on both ARPES measurements, which observed cylindrical light electron bands~\\cite{FujimoriJPSJ2019,MiaoPRL2020} and recent de Haas\u2013van Alphen measurements that reveal heavy cylindrical bands ~\\cite{Aoki2022First}. \n\nThe isotropic fully gapped model $A_{1u}$ saturates at $T\/T_{\\rm c}=0.2$ [Fig.~S6]. In contrast, the experimental data saturate at a lower temperature, which implies an anisotropic structure in the SC gap function. The calculated normalized superfluid density $n_{(011)}\\sim(n_a+n_{\\perp[011],a})\/2$ for highly anisotropic $A_{1u}$ and $B_{1u}$ have a similar temperature dependence compared to our experimental results, whereas $n_{(011)}$ for $B_{2u}$ and $B_{3u}$ do not agree with our data because of their point nodes near the (011) plane with isotropic or ellipsoidal 3D Fermi surfaces [Figs.~4(a),4(b)]. For highly anisotropic $A_{1u}$ and $B_{3u}$, $n_{(011)}$ agrees with our experimental results, whereas $n_{(011)}$ for $B_{2u}$ inconsistent with the data for a cylindrical Fermi surface [Fig.~4(c)]. We note that our calculations with point nodes do not completely explain our results near zero temperature, which may be caused by our assumptions of the simplest structures of Fermi surface and gap functions, the simple averaging of $n_a$ and $n_{\\perp[011],a}$, or by the assumption of a single band.\nOur results indicate the existence of point nodes along the $a$ axis for a cylindrical Fermi surface. A highly anisotropic fully-gapped component is also allowed.\n\n\nIn summary, we microscopically imaged the superfluid density and the vortex density in high quality samples of UTe$_2$. The superfluid density is homogeneous, and the temperature dependence below the SC transition $T_{\\rm c}$ does not show evidence for a second phase transition. The observed temperature dependence of the superfluid density can be explained by a $B_{1u}$ order parameter for a 3D ellipsoidal (or isotropic) Fermi surfaces or by a $B_{3u}$ order parameter for a quasi-2D cylindrical Fermi surface. A highly anisotropic $A_{1u}$ symmetry component is also allowed for any Fermi surface structures. Combining our results with previous studies about the gap symmetry of UTe$_2$~\\cite{Bae2021Ano,Metz2019Point,Kittaka2020Ori,Fujibayashi2022Super}, we conclude that the SC order parameter is most likely dominated by the $B_{3u}$ symmetry. In light of our results, evidence for time-reversal symmetry breaking and chiral superconductivity in UTe$_2$ could be understood either through the presence of vortices and antivortices even at zero applied field or by the presence of a finite anisotropic $A_{1u}$ symmetry in the SC order parameter. \n\n\n\n\n\\begin{acknowledgments}\nThe authors thank J. Ishizuka for fruitful discussions. This work was primarily supported by the Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract No. DE- AC02-76SF00515. Sample synthesis at LANL was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering \"Quantum Fluctuations in Narrow-Band Systems\" project, while heat capacity measurements were performed with support from the LANL LDRD program. Y.I. was supported by the Japan Society for the Promotion of Science (JSPS), Overseas Research Fellowship.\n\\end{acknowledgments}\n\n\n\n\t\n\t\n\\section*{Contributions}\nY.I. carried out the scanning SQUID microscopy, analyzed experimental data, simulated the superfluid density, and wrote the manuscript. H.M. carried out the scanning SQUID microscopy. S.M.T., F.R., and P.F.S.R. synthesized the crystals. K.A.M. supervised the project. All the authors discussed the results and implications and commented on the manuscript.\n\t\n\t\n\\section*{Additional information}\nCorrespondence and requests for materials should be addressed to Y. I. (yiguchi@stanford.edu)\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nConsider the problem of finding a local minimizer of the expectation\n$F(x)\\eqdef\\bE(f(x,\\xi))$ w.r.t. $x\\in \\bR^d$, where $f(\\,.\\,,\\xi)$ is\na possibly non-convex function depending on some random\nvariable~$\\xi$.\nThe distribution of $\\xi$ is assumed unknown, but revealed online by\nthe observation of iid copies $(\\xi_n:n\\geq 1)$ of the r.v.~$\\xi$.\nStochastic gradient descent (SGD) is the most classical algorithm\nto search for such a minimizer.\nVariants of SGD which include an inertial term have also become very popular.\nIn these methods, the update rule depends on a parameter called\nthe \\emph{learning rate}, which is generally assumed constant or vanishing.\nThese algorithms, although widely used, have at least two limitations.\nFirst, the choice of the learning rate is generally difficult; large learning rates result in\nlarge fluctuations of the estimate, whereas small learning rates\ninduce slow convergence.  Second, a common learning rate is used for\nevery coordinate despite the possible discrepancies in the values of\nthe gradient vector's coordinates.\n\nTo alleviate these limitations, the popular\n\\adam\\ algorithm \\cite{kingma2014adam} adjusts the learning rate\ncoordinate-wise, as a function of the past values of the squared\ngradient vectors' coordinates. The algorithm thus combines the assets\nof inertial methods with an adaptive per-coordinate learning rate\nselection. Finally, the algorithm includes a so-called\n\\emph{bias correction} step. Acting on the current estimate of the gradient vector,\nthis step is especially useful during the early iterations.\n\nDespite the growing popularity of the algorithm, only few works\ninvestigate its behavior from a theoretical point of\nview (see the discussion in Section~\\ref{sec:related-works}).\nThe present paper studies the convergence of \\adam\\ from a dynamical system viewpoint.\\\\\n\n\\noindent{\\bf Contributions}\n\\begin{itemize}[leftmargin=*]\n\\item We introduce a continuous-time version of the \\adam\\ algorithm under the form\n  of a non-autonomous ordinary differential equation (ODE).\n  Building on the existence of an explicit Lyapunov function for the ODE,\n  we show the existence of a unique global solution to the ODE. This first result\n  turns out to be non-trivial due to the irregularity of the vector field.\n  We then establish the convergence of the continuous-time \\adam\\\n  trajectory to the set of critical points of the objective function $F$.\nThe proposed continuous-time version of \\adam\\ provides useful\n  insights on the effect of the bias correction step.  It is shown\n  that, close to the origin, the objective function~$F$ is\n  non-increasing along the \\adam\\ trajectory,\n  suggesting that early iterations of \\adam\\ can only improve the initial guess.\n\n\\item Under a \\L{}ojasiewicz-type condition,\n  we prove that the solution to the ODE converges to a single critical point of the objective\n  function $F$. We provide convergence rates in this case.\n\n\\item\n  In discrete time, we first analyze the \\adam\\ iterates in the constant stepsize\n  regime as originally introduced in \\cite{kingma2014adam}.\n  In this work, it is shown that the discrete-time \\adam\\ iterates shadow the\n  behavior of the non-autonomous ODE in the asymptotic regime where\n  the stepsize parameter $\\gamma$ of \\adam\\ is small.  More\n  precisely, we consider the interpolated process $\\sz^\\gamma(t)$\n  which consists of a piecewise linear interpolation of the \\adam\\ iterates.\n  The random process $\\sz^\\gamma$ is indexed by the parameter~$\\gamma$, which is\n  assumed constant during the whole run of the algorithm.\n  In the space of continuous functions on $[0,+\\infty)$ equipped\n  with the topology of uniform convergence on compact sets,\n  we establish that  $\\sz^\\gamma$ converges in probability\n  to the solution to the non-autonomous ODE when $\\gamma$ tends to zero.\n\n\\item Under a stability condition, we prove the asymptotic ergodic convergence\n  of the probability of the discrete-time \\adam\\ iterates to approach the set of critical\n  points of the objective function in the doubly asymptotic regime\n  where $n\\to\\infty$ then $\\gamma\\to 0$.\n\\item\nBeyond the original constant stepsize \\adam\\,, we propose\na decreasing stepsize version of the algorithm.\nWe provide sufficient conditions ensuring the stability and the almost sure convergence of the iterates towards\nthe critical points of the objective function.\n\\item  We establish a convergence rate of the stochastic iterates of the decreasing stepsize algorithm under the\nform of a conditional central limit theorem.\n\n\\end{itemize}\nWe claim that our analysis can be easily extended to\nother adaptive algorithms such as e.g. \\textsc{RmsProp} or\n\\textsc{AdaGrad} \\cite{tieleman2012lecture,duchi2011adaptive}\nand \\textsc{AmsGrad} (see Section~\\ref{sec:related-works}).\\\\%\\medskip\n\nThe paper is organized as follows.\nIn Section~\\ref{sec:adam}, we present\nthe \\adam\\ algorithm and the main assumptions.\nOur main results are stated in Sections~\\ref{sec:continuous_time} to \\ref{sec:discrete_decreasing}.\nWe provide a review of related works in Section~\\ref{sec:related-works}.\nThe rest of the paper addresses the proofs of our results\n(Sections~\\ref{sec:proofs_cont_time} to \\ref{sec:proofs_sec_discrete_decreasing}).\n\\hfill\\\\\n\n\\noindent{\\bf Notations}. If $x$, $y$ are two vectors on $\\bR^d$ for some $d\\geq 1$, we denote by $x \\odot y$, $x^{\\odot 2}$, $x\/y$, $|x|$, $\\sqrt{|x|}$ the vectors\non $\\bR^d$ whose $i$-th coordinates are respectively given by $x_iy_i$, $x_i^2$, $x_i\/y_i$, $|x_i|$, $\\sqrt{|x_i|}$.\nInequalities of the form $x\\leq y$ are read componentwise.\nDenote by $\\|\\cdot\\|$ the standard Euclidean norm.\nFor any vector $v\\in (0,+\\infty)^d$, write $\\|x\\|^2_v = \\sum_i v_i x_i^2$.\nNotation $A^T$ represents the transpose of a matrix $A$.\nIf  $z\\in \\bR^d$ and $A$ is a non-empty subset of $\\bR^d$,\nwe use the notation $\\mathsf d(z,A) \\eqdef \\inf\\{ \\|z-z'\\| :z'\\in A\\}$.\nIf $A$ is a set, we denote  by $\\1_A$ the function equal to one on that set and to zero elsewhere.\n We denote by $C([0,+\\infty),\\bR^d)$ the space of continuous functions from $[0,+\\infty)$ to $\\bR^d$ endowed with the topology of\nuniform convergence on compact intervals.\n\n\\section{The \\adam\\ Algorithm}\n\\label{sec:adam}\n\n\\subsection{Algorithm and Assumptions}\n\nLet $(\\Omega,\\cF,\\bP)$ be a probability space, and let\n$(\\Xi,\\mathfrak{S})$ denote an other measurable space.  Consider a\nmeasurable map $f:\\bR^d\\times \\Xi\\to \\bR$, where $d$ is an integer.\nFor a fixed value of $\\xi$, the mapping $x\\mapsto f(x,\\xi)$ is\nsupposed to be differentiable, and its gradient w.r.t. $x$ is denoted\nby $\\nabla f(x,\\xi)$.  Define $\\cZ\\eqdef \\bR^d\\times\\bR^d\\times\n\\bR^d$, $\\cZ_{+}\\eqdef \\bR^d\\times\\bR^d\\times [0,+\\infty)^d$ and\n$\\cZ_{+}^*\\eqdef \\bR^d\\times\\bR^d\\times (0,+\\infty)^d$. \\adam\\\ngenerates a sequence $z_n\\eqdef (x_n,m_n,v_n)$ on\n$\\cZ_+$ given by Algorithm~\\ref{alg:adam}.\n\\begin{algorithm}[tb]\n   \\caption{\\bf \\adam$(\\gamma,\\alpha,\\beta,\\varepsilon)$.}\n   \\label{alg:adam}\n\\begin{algorithmic}\n   \\STATE {\\bfseries Initialization:} $x_0\\in \\bR^d, m_0=0$, $v_0=0$.\n   \\FOR{$n=1$ {\\bfseries to} $n_{\\text{iter}}$}\n   \\STATE $m_n = \\alpha m_{n-1} + (1-\\alpha)  \\nabla f(x_{n-1},\\xi_n)$\n   \\STATE $v_n = \\beta v_{n-1} + (1-\\beta) \\nabla f(x_{n-1},\\xi_n)^{\\odot 2}$\n   \\STATE $\\hat m_{n} = m_{n}\/(1-\\alpha^{n})$\n   \\COMMENT{bias correction step}\n   \\STATE $\\hat v_{n} = v_{n}\/(1-\\beta^{n})$\n   \\COMMENT{bias correction step}\n   \\STATE $x_{n} = x_{n-1} - \\gamma \\hat m_{n} \/ (\\varepsilon+\\sqrt{\\hat v_{n}}) \\,.$\n   \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\nIt satisfies:\n$\nz_n=T_{\\gamma,\\alpha,\\beta}(n,z_{n-1},\\xi_n)\\,,\n$\nfor every $n\\geq 1$, where for every $z=(x,m,v)$ in $\\cZ_+$, $\\xi\\in \\Xi$,\n\\begin{equation}\n  T_{\\gamma,\\alpha,\\beta}(n,z,\\xi) \\eqdef\n  \\begin{pmatrix}\n x -\\frac{\\gamma (1-\\alpha^{n})^{-1}(\\alpha m+(1-\\alpha)\\nabla f(x,\\xi))}{ \\varepsilon+{(1-\\beta^{n})^{-1\/2} (\\beta v+(1-\\beta)\\nabla f(x,\\xi)^{\\odot 2})^{1\/2}}} \\\\\n\\alpha m + (1-\\alpha) \\nabla f(x,\\xi) \\\\\n\\beta v + (1-\\beta) \\nabla f(x,\\xi)^{\\odot 2}\n\\end{pmatrix}\\,.\\label{eq:T}\n\\end{equation}\n\n\\begin{remark}\n\\label{rem:debiasing}\n\\quad\\quad The iterates $z_n$ form a non-homogeneous Markov chain, because $T_{\\gamma,\\alpha,\\beta}(n,z,\\xi)$ depends on $n$.\nThis is due to the so-called debiasing step, which consists of replacing $m_n,v_n$ in Algorithm~\\ref{alg:adam}\nby their ``debiased'' versions $\\hat m_n,\\hat v_n$. The motivation becomes clear when expanding the expression:\n\\begin{equation*}\n\\hat m_n = \\frac{m_n}{1-\\alpha^n} = \\frac {1-\\alpha}{1-\\alpha^n}\\sum_{k=0}^{n-1}\\alpha^k \\nabla f(x_k,\\xi_{k+1})\\,. \\label{eq:debiasing}\n\\end{equation*}\nFrom this equation, it is observed that,  $\\hat m_n$ forms a convex combination of the past gradients.\nThis is unlike $m_n$, which may be small during the first iterations.\n\n\\end{remark}\n\n\\begin{assumption} \\label{hyp:model}\nThe mapping $f:\\bR^d\\times\\Xi\\to \\bR$ satisfies the following.\n  \\begin{enumerate}[{\\sl i)}]\n  \\item For every $x\\in \\bR^d$, $f(x,\\,.\\,)$ is $\\mathfrak{S}$-measurable.\n  \\item For almost every $\\xi$, the map $f(\\,.\\,,\\xi)$ is continuously differentiable.  \\label{hyp:dif}\n  \\item There exists $x_*\\in \\bR^d$ s.t. $\\bE(|f(x_*,\\xi)|)<\\infty$ and  $\\bE(\\|\\nabla f(x_*,\\xi)\\|^2)<\\infty$.\n  \\item For every compact subset $K\\subset \\bR^d$, there exists $L_K>~0$ such that\n   for every $(x,y)\\in K^2$, $\\bE(\\|\\nabla f(x,\\xi)-\\nabla f(y,\\xi)\\|^2)\\leq L_K^2\\|x-y\\|^2$.\n  \\end{enumerate}\n\\end{assumption}\n\nUnder Assumption~\\ref{hyp:model}, it is an easy exercise to show that the mappings\n$F:\\bR^d\\to\\bR$ and $S:\\bR^d\\to\\bR^d$, given by:\n\\begin{equation}\n  F(x) \\eqdef \\bE(f(x,\\xi)) \\quad \\text{and} \\quad S(x) \\eqdef \\bE(\\nabla f(x,\\xi)^{\\odot 2})\\label{eq:F_and_S}\n\\end{equation}\nare well defined; $F$ is continuously differentiable and by\nLebesgue's dominated convergence theorem, $\\nabla F(x) = \\bE(\\nabla\nf(x,\\xi))$ for all $x$. Moreover, $\\nabla F$ and $S$ are locally Lipschitz continuous.\n\\begin{assumption}\n  \\label{hyp:coercive}\n$F$ is coercive.\n\\end{assumption}\n\\begin{assumption}\n  \\label{hyp:S>0}\nFor every $x\\in \\bR^d$, $S(x)>0$.\n\\end{assumption}\nIt follows from our assumptions that the set of critical points of\n$F$, denoted by $$\\cS \\eqdef \\nabla F^{-1}(\\{0\\}),$$ is non-empty.\nAssumption~\\ref{hyp:S>0} means that there is \\emph{no} point $x\\in\n\\bR^d$ satisfying $\\nabla f(x,\\xi) = 0$ with probability one (w.p.1). This is\na mild hypothesis in practice.\n\n\\subsection{Asymptotic Regime}\n\nWe address the constant stepsize regime, where\n$\\gamma$ is fixed along the iterations (the default value recommended\nin \\cite{kingma2014adam} is $\\gamma = 0.001$). As opposed to the decreasing stepsize context,\nthe sequence $z_n^\\gamma\\eqdef z_n$ \\emph{cannot} in general converge as $n$ tends to infinity, in an almost sure sense.\nInstead, we investigate the asymptotic behavior of the {family} of processes $(n\\mapsto z_n^\\gamma)_{\\gamma>0}$\nindexed by $\\gamma$, in the regime where $\\gamma\\to0$. We use the so-called ODE method (see e.g. \\cite{ben-(cours)99}).\nThe interpolated process $\\sz^{\\gamma}$ is the\npiecewise linear function defined on $[0,+\\infty)\\to \\cZ_+$\nfor all $t \\in [n\\gamma ,(n+1)\\gamma)$  by:\n\\begin{equation}\n\\sz^{ \\gamma}(t) \\eqdef z_{n}^{ \\gamma} + (z_{n+1}^{ \\gamma}-z_{n}^{ \\gamma})\\left(\\frac {t-n\\gamma}{\\gamma}\\right)\\,.\n\\label{eq:interpolated-process}\n\\end{equation}\nWe establish the convergence in probability of the\nfamily of random processes $(\\sz^{ \\gamma})_{\\gamma>0}$ as $\\gamma$ tends to zero, towards a\ndeterministic continuous-time system defined by an ODE.\nThe latter ODE, which we provide below at Eq.~(\\ref{eq:ode}), will be referred to\nas the continuous-time version of \\adam.\n\nBefore describing the ODE, we need to be more specific about our\nasymptotic regime. As opposed to SGD, \\adam\\ depends on\ntwo parameters $\\alpha$, $\\beta$, in addition to the stepsize $\\gamma$.\nThe  paper \\cite{kingma2014adam} recommends choosing the constants $\\alpha$ and $\\beta$ close to one\n(the default values $\\alpha=0.9$ and $\\beta=0.999$ are suggested).\nIt is thus legitimate to assume that\n$\\alpha$ and $\\beta$ tend to one, as $\\gamma$ tends to zero.\nWe set  $\\alpha \\eqdef \\bar \\alpha(\\gamma)$ and $\\beta\\eqdef\\bar\\beta(\\gamma)$, where\n$\\bar \\alpha(\\gamma)$ and $\\bar\\beta(\\gamma)$ converge to one as $\\gamma\\to 0$.\n\\begin{assumption}\n\\label{hyp:alpha-beta}\nThe functions $\\bar \\alpha:\\bR_+\\to [0,1)$ and $\\bar \\beta:\\bR_+\\to [0,1)$ are\ns.t. the following limits exist:\n  \\begin{equation}\n    a\\eqdef \\lim_{\\gamma_\\downarrow 0} \\frac{1-\\bar\\alpha(\\gamma)}\\gamma ,\\quad b\\eqdef  \\lim_{\\gamma_\\downarrow 0}\\frac{1-\\bar\\beta(\\gamma)}\\gamma\\,.\\label{eq:regime}\n  \\end{equation}\nMoreover, $a>0$, $b>0$, and the following condition holds: $b\\leq 4a\\,.$\n\\end{assumption}\nNote that the condition $b\\leq 4a$ is compatible with the default settings recommended\nby \\cite{kingma2014adam}.\nIn our model, we shall now replace the map $T_{\\gamma,\\alpha,\\beta}$ by $T_{\\gamma,\\bar \\alpha(\\gamma),\\bar \\beta(\\gamma)}$.\nLet $x_0\\in \\bR^d$ be fixed. For any fixed $\\gamma>0$, we define the sequence $(z_n^\\gamma)$ generated by\n\\adam\\ with a fixed stepsize~$\\gamma>0$:\n\\begin{equation}\nz_n^\\gamma \\eqdef T_{\\gamma,\\bar \\alpha(\\gamma),\\bar \\beta(\\gamma)}(n,z^\\gamma_{n-1},\\xi_n)\\,,\\label{eq:znT}\n\\end{equation}\nthe initialization being chosen as $z_0^\\gamma=(x_0,0,0)$.\n\n\n\\section{Continuous-Time System}\n\\label{sec:continuous_time}\n\n\\subsection{Ordinary Differential Equation}\n\nIn order to gain insight into the behavior of the sequence $(z_n^\\gamma)$\ndefined by (\\ref{eq:znT}),\nit is convenient to rewrite the \\adam\\ iterations under the following equivalent form, for every $n\\geq 1$:\n\\begin{equation}\nz_n^{\\gamma} = z^{\\gamma}_{n-1} + \\gamma h_{\\gamma}(n,z^{\\gamma}_{n-1}) + \\gamma \\Delta^\\gamma_{n}\\,,\\label{eq:RM}\n\\end{equation}\nwhere we define for every $\\gamma>0$, $z\\in \\cZ_+$,\n\\begin{equation}\nh_\\gamma(n,z) \\eqdef \\gamma^{-1}\\bE(T_{\\gamma,\\bar\\alpha(\\gamma),\\bar\\beta(\\gamma)}(n,z,\\xi)-z)\\,,\\label{eq:hgamma}\n\\end{equation}\nand where $\\Delta^\\gamma_{n}\\eqdef \\gamma^{-1}(z_n^{\\gamma} - z^{\\gamma}_{n-1}) - h_{\\gamma}(n,z^{\\gamma}_{n-1})$.\nNote that $(\\Delta^\\gamma_{n})$ is a martingale increment noise sequence in the sense that\n$\\bE(\\Delta^\\gamma_{n}|\\cF_{n-1}) = 0$ for all $n\\geq 1$, where $\\cF_n$ stands for the $\\sigma$-algebra generated\nby the r.v. $\\xi_1,\\dots,\\xi_n$.\nDefine the map $h:(0,+\\infty)\\times \\cZ_+\\to\\cZ$ for all $t>0$, all $z=(x,m,v)$ in $\\cZ_+$ by:\n\\begin{equation}\n  \\label{eq:h}\n  h(t,z)= \\begin{pmatrix}\n  -\\frac{(1-e^{-at})^{-1}m}{\\varepsilon+\\sqrt{(1-e^{-bt})^{-1} v}} \\\\\na (\\nabla F(x)-m) \\\\\nb (S(x)-v)\n\\end{pmatrix} \\,,\n\\end{equation}\nwhere $a,b$ are the constants defined in Assumption~\\ref{hyp:alpha-beta}.\nWe prove that, for any fixed $(t,z)$, the quantity $h(t,z)$ coincides with the limit\nof $h_\\gamma(\\lfloor t\/\\gamma\\rfloor,z)$ as $\\gamma\\downarrow 0$. This remark along with Eq.~(\\ref{eq:RM})\nsuggests that, as $\\gamma\\downarrow 0$, the interpolated process $\\sz^\\gamma$ shadows the non-autonomous differential equation\n\\begin{equation}\n  \\begin{array}[h]{l}\n\\dot z(t) = h(t, z(t))\\,.\n\\end{array}\n\\tag{ODE}\n\\label{eq:ode}\n\\end{equation}\n\n\\subsection{Existence, Uniqueness, Convergence}\n\\label{subsec:odeanalysis}\n\nSince $h(\\,.\\,,z)$ is non-continuous\nat point zero for a fixed $z\\in \\cZ_+$, and since $h(t,\\,.\\,)$\nis not locally Lipschitz continuous for a fixed~$t~>~0$,\nthe existence and uniqueness of the solution to (\\ref{eq:ode}) do not stem directly from off-the-shelf theorems.\n\nLet {\\color{black}$x_0$ be fixed}.\nA continuous map $z:[0,+\\infty)\\to\\cZ_+$ is said to be a global solution to (\\ref{eq:ode}) with initial condition {\\color{black}$(x_0,0,0)$}\nif $z$ is continuously differentiable on $(0,+\\infty)$, if Eq.~(\\ref{eq:ode})\nholds for all $t>0$, and if {\\color{black}$z(0)=(x_0,0,0)$}.\n\n\\begin{theorem}[Existence and uniqueness]\n  \\label{th:exist-unique}\n \n  Let Assumptions~\\ref{hyp:model} to \\ref{hyp:alpha-beta} hold true.\n  There exists a unique global solution $z:[0,+\\infty)\\to\\cZ_+$\n  to~(\\ref{eq:ode}) with\n  initial condition $(x_0,0,0)$.\n  Moreover, $z([0,+\\infty))$ is a bounded subset of $\\cZ_+$.\n\\end{theorem}\nOn the other hand, we note that a solution may not exist for\nan initial point$(x_0,m_0,v_0)$ with arbitrary (non-zero) values of $m_0, v_0$.\n\n\\begin{theorem}[Convergence]\n\\label{th:cv-adam}\nLet Assumptions~\\ref{hyp:model} to \\ref{hyp:alpha-beta} hold true.\nAssume that $F(\\cS)$ has an empty interior.\nLet $z:t\\mapsto (x(t),m(t),v(t))$ be\nthe global solution to~(\\ref{eq:ode}) with the initial condition\n$(x_0,0,0)$.\\,\nThen, the set $\\cS$ is non-empty and $\\lim_{t\\to\\infty} \\sd(x(t),\\cS) =0$,\n $\\lim_{t\\to\\infty}m(t)=0$, $\\lim_{t\\to\\infty}S(x(t))-v(t)=0$.\n\n\n\\end{theorem}\n\n\\noindent{\\bf Lyapunov function.} The proof of Th.~\\ref{th:exist-unique}\n  relies on the existence of a Lyapunov function for the non-autonomous\n  equation~(\\ref{eq:ode}).  Define  $V:(0,+\\infty)\\times \\cZ_+\\to \\bR$ by\n  \\begin{equation}\n    \\label{eq:V}\n    V(t,z)\\eqdef F(x)+\\frac 12 \\left\\|m\\right\\|^2_{U(t,v)^{-1}}\\,,\n  \\end{equation}\n  for every $t>0$ and every $z=(x,m,v)$ in $\\cZ_+$, where\n  $U:(0,+\\infty)\\times [0,+\\infty)^d\\to \\bR^d$ is the map given by:\n  \\begin{equation}\n    \\label{eq:U}\n    U(t,v) \\eqdef a(1-e^{-at})\\left(\\varepsilon+\\sqrt{\\frac{v}{1-e^{-bt}}}\\right)\\,.\n  \\end{equation}\nThen, $t\\mapsto V(t,z(t))$ is decreasing if $z(\\cdot)$ is the global solution to~(\\ref{eq:ode}).\n\n\\noindent{\\bf Cost decrease at the origin.} As $F$ itself is not a Lyapunov function for~(\\ref{eq:ode}),\nthere is no guarantee that $F(x(t))$ is decreasing w.r.t. $t$.\nNevertheless, the statement holds at the origin. Indeed, it can be shown that\n$\\lim_{t\\downarrow 0}V(t,z(t))=F(x_0)$ (see Prop.~\\ref{prop:adam-bounded}). As a consequence,\n\\begin{equation}\n  \\forall t\\geq 0,\\ F(x(t))\\leq F(x_0)\\,.\n\\label{eq:costdecrease}\n\\end{equation}\n\nIn other words, the (continuous-time) \\adam\\ procedure\n\\emph{can only improve} the initial guess $x_0$.\nThis is the consequence of the so-called bias correction steps in \\adam\\ (see Algorithm~\\ref{alg:adam})\\,.\nIf these debiasing steps were deleted in the \\adam\\ iterations,\nthe early stages of the algorithm could degrade the initial estimate $x_0$.\n\n\\noindent{\\bf Derivatives at the origin.}\nThe proof of Th.~\\ref{th:exist-unique} reveals that the initial derivative is given\nby $\\dot x(0) = -\\nabla F(x_0)\/(\\varepsilon+\\sqrt{S(x_0)})$ (see Lemma~\\ref{lem:m-v-derivables-en-zero}).\nIn the absence of debiasing steps, the initial derivative $\\dot x(0)$ would be a function of the initial\nparameters $m_0$, $v_0$, and the user would be required to tune these hyperparameters.\nNo such tuning is required thanks to the debiasing step.\nWhen $\\varepsilon$ is small and when the variance of $\\nabla f(x_0,\\xi)$ is small (\\emph{i.e.}, $S(x_0)\\simeq \\nabla F(x_0)^{\\odot 2}$),\nthe initial derivative $\\dot x(0)$ is approximately equal to $-\\nabla F(x_0)\/|\\nabla F(x_0)|$.\nThis suggests that in the early stages of the algorithm, the \\adam\\ iterations\nare comparable to the \\emph{sign} variant of the gradient descent, the properties of which were\ndiscussed in previous works, see \\cite{balles2018dissecting}\n\n\\subsection{Convergence rates}\n\nIn this paragraph, we establish the convergence to a single critical\npoint of $F$ and quantify the convergence rate, using the following\nassumption \\cite{lojasiewicz1963}.\n\\begin{assumption}[\\L{}ojasiewicz property]\n  \\label{hyp:lojasiewicz_prop}\nFor any  $x^* \\in {\\mathcal S}$, there exist\n$c >0\\,, \\sigma >0$ and $\\theta \\in (0,\\frac 12]$ s.t.\n\\begin{equation}\n  \\label{eq:lojasiewicz}\n\\forall x \\in \\bR^d \\,\\,\\text{s.t}\\,\\,\\|x-x^*\\|\\leq \\sigma\\,,\\quad  \\|\\nabla F(x)\\| \\geq c |F(x) - F(x^*)|^{1-\\theta}\\,.\n\\end{equation}\n\\end{assumption}\nAssumption~\\ref{hyp:lojasiewicz_prop} holds for real-analytic functions\nand semialgebraic functions.\nWe refer to\n\\cite{harauxjendoubi2015,attouch2009convergence,bolte2014proximal}\nfor a discussion and a review of applications.\nWe will call any $\\theta$ satisfying (\\ref{eq:lojasiewicz}) for some $c,\\sigma>0$,\nas a \\L{}ojasiewicz exponent of $f$ at $x^*$. The next result establishes the convergence\nof the function $x(t)$ generated by the ODE to a single critical point of $f$,\nand provides the convergence rate as a function of the \\L{}ojasiewicz exponent of $f$\nat this critical point. The proof is provided in subsection~\\ref{sec:cont_asymptotic_rates}.\n\\begin{theorem}\n  \\label{thm:asymptotic_rates}\n \n  Let Assumptions~\\ref{hyp:model} to \\ref{hyp:alpha-beta} and \\ref{hyp:lojasiewicz_prop} hold true.\n  Assume that $F(\\cS)$ has an empty interior. Let $x_0 \\in \\bR^d$ and let $z:t\\mapsto (x(t),m(t),v(t))$ be\n  the global solution to~(\\ref{eq:ode}) with initial condition $(x_0,0,0)$.\n  Then, there exists $x^* \\in \\cS$ such that $x(t)$ converges to $x^*$ as $t \\to +\\infty$.\n\n  Moreover, if $\\theta\\in (0,\\frac 12]$ is a \\L{}ojasiewicz exponent of $f$ at $x^*$,\n there exists a constant $C >0$ s.t. for all $t\\geq 0$,\n  \\begin{align*}\n    \\|x(t)-x^*\\| &\\leq C t^{-\\frac{\\theta}{1-2\\theta}}\\,,\\quad \\text{if}\\,\\,\\, 0<\\theta<\\frac{1}{2}\\,,\\\\\n    \\|x(t)-x^*\\| &\\leq C e^{- \\delta t} \\,,\\quad \\text{for some}\\,\\, \\delta >0\\,\\,\\text{if}\\,\\, \\theta = \\frac 12 \\,.\n  \\end{align*}\n\\end{theorem}\n\n\\section{Discrete-Time System: Convergence of \\adam}\n\\label{sec:discrete}\n\n\\begin{assumption}\n  \\label{hyp:iid}\nThe sequence $(\\xi_n:n\\geq 1)$ is iid, with the same distribution as~$\\xi$.\n\\end{assumption}\n\n\\begin{assumption}\n  \\label{hyp:moment-f}\nLet $p>0$. Assume either one of the following conditions.\n\\begin{enumerate}[i)]\n\\item \\label{momentegal} For every compact set $K\\subset \\bR^d$, $\\sup_{x\\in K} \\bE(\\|\\nabla f(x,\\xi)\\|^{p})<\\infty\\,.$\n\\item \\label{momentreinforce} For every compact set $K\\subset \\bR^d$,  $\\exists\\, p_K>p$, $\\sup_{x\\in K} \\bE(\\|\\nabla f(x,\\xi)\\|^{p_K})<\\infty.$\n\\end{enumerate}\n\\end{assumption}\nThe value of $p$ will be specified in the sequel, in the statement of\nthe results. Clearly, Assumption~\\ref{hyp:moment-f}~\\ref{momentreinforce} is stronger than Assumption~\\ref{hyp:moment-f}~\\ref{momentegal}.\nWe shall use either the latter or the former in our statements.\n\n\\begin{theorem}\n  \\label{th:weak-cv}\nLet Assumptions~\\ref{hyp:model} to \\ref{hyp:alpha-beta} and \\ref{hyp:iid} hold true. Let Assumption~\\ref{hyp:moment-f}~\\ref{momentreinforce}\nhold with $p=2$.\nConsider $x_0\\in \\bR^d$. For every $\\gamma>0$, let $(z_n^\\gamma:n\\in \\bN)$ be the random sequence defined by the \\adam\\ iterations~(\\ref{eq:znT})\nand $z_0^\\gamma = (x_0,0,0)$. Let $\\sz^\\gamma$ be the corresponding interpolated process defined by Eq.~(\\ref{eq:interpolated-process}).\nFinally, let $z$ denote the unique global solution to (\\ref{eq:ode}) issued from $(x_0,0,0)$.\nThen,\n$$\n\\forall\\, T>0,\\ \\forall\\, \\delta>0,\\ \\ \\lim_{\\gamma\\downarrow 0}\\bP\\left(\\sup_{t\\in[0,T]}\\|\\sz^\\gamma(t)-z(t)\\|>\\delta\\right)=0\\,.\n$$\n\\end{theorem}\nRecall that a family of r.v. $(X_\\alpha)_{\\alpha\\in I}$ is called \\emph{bounded in probability}, or \\emph{tight}, if for every\n$\\delta>0$, there exists a compact set $K$ s.t. $\\bP(X_\\alpha\\in K)\\geq 1-\\delta$ for every $\\alpha\\in I$.\n\\begin{assumption}\n\\label{hyp:tight}\n  There exists $\\bar \\gamma_0>0$ s.t. the family of r.v. $(z_n^\\gamma:n\\in \\bN,0<\\gamma<\\bar \\gamma_0)$ is bounded in probability.\n\\end{assumption}\n\n\\begin{theorem}\nConsider $x_0\\in \\bR^d$. For every $\\gamma>0$, let $(z_n^\\gamma:n\\in \\bN)$ be the random sequence defined by the \\adam\\ iterations~(\\ref{eq:znT})\nand $z_0^\\gamma = (x_0,0,0)$.\nLet Assumptions~\\ref{hyp:model} to \\ref{hyp:alpha-beta}, \\ref{hyp:iid} and \\ref{hyp:tight} hold.\nLet Assumption~\\ref{hyp:moment-f}~\\ref{momentreinforce} hold with $p=2$. Then,\nfor every $\\delta > 0$,\n  \\begin{equation}\n    \\label{eq:long-run}\n    \\lim_{\\gamma\\downarrow 0}\\limsup_{n\\to\\infty} \\frac 1{n} \\sum_{k=1}^n \\bP(\\sd(x_k^\\gamma, \\cS)>\\delta) = 0\\,.\n  \\end{equation}\n\\label{th:longrun}\n\\end{theorem}\n\n\n\\noindent{\\bf Convergence in the long run.}\nWhen the stepsize $\\gamma$ is constant, the sequence\n$(x_n^\\gamma)$ cannot converge in the almost sure sense as\n$n\\to\\infty$.\nConvergence may only hold in the doubly asymptotic regime where\n$n\\to\\infty$ then $\\gamma\\to 0$.\n\n\\noindent{\\bf Randomization.} For every $n$, consider a r.v. $N_n$  uniformly\ndistributed on  $\\{1,\\dots,n\\}$. Define $\\tilde x_n^\\gamma = x_{N_n}^\\gamma$.\nWe obtain from Th.~\\ref{th:longrun} that for every $\\delta>0$,\n$$\n\\limsup_{n\\to\\infty} \\ \\bP(\\sd(\\tilde x_n^\\gamma, \\cS)>\\delta) \\xrightarrow[\\gamma\\downarrow 0]{} 0\\,.\n$$\n\n\\noindent{\\bf Relationship between discrete and continuous time \\adam.}\nTh.~\\ref{th:weak-cv} means that the family of random processes $(\\sz^\\gamma:~\\gamma>0)$ converges\nin probability as $\\gamma\\downarrow 0$ towards the unique solution to~(\\ref{eq:ode}) issued from $(x_0,0,0)$.\nThis motivates the fact that the non-autonomous system~(\\ref{eq:ode})\nis a relevant approximation to the behavior of the iterates $(z_n^\\gamma:n\\in \\bN)$ for a small\nvalue of the stepsize~$\\gamma$.\n\n\\noindent{\\bf Stability.} Assumption~\\ref{hyp:tight} ensures\nthat the iterates $z_n^\\gamma$ do not explode in the long run.\nA sufficient condition is for instance that\n$\\sup_{n,\\gamma} \\bE \\|z_n^\\gamma\\|<\\infty\\,.$\n{In theory, this assumption can be difficult to verify.}\nNevertheless, in practice, a projection step on a compact set\ncan be introduced to ensure the boundedness of the estimates.\n\n\\section{A Decreasing Stepsize \\adam\\, Algorithm}\n\\label{sec:discrete_decreasing}\n\n\\subsection{Algorithm}\n\n\\adam\\, inherently uses constant stepsizes. Consequently, the\niterates~(\\ref{eq:znT}) do not converge in the almost sure sense.\nIn order to achieve convergence, we introduce in this section a decreasing\nstepsize version of \\adam. The iterations are given in Algorithm~\\ref{alg:adam-decreasing}.\n\\begin{algorithm}[tb]\n   \\caption{\\bf \\adam - decreasing stepsize $(((\\gamma_n,\\alpha_n,\\beta_n):n\\in \\bN^*), \\varepsilon)$.}\n   \\label{alg:adam-decreasing}\n\\begin{algorithmic}\n  \\STATE {\\bfseries Initialization:} $x_0\\in \\bR^d, m_0=0$, $v_0=0$, $r_0=\\bar r_0=0$.\n   \\FOR{$n=1$ {\\bfseries to} $n_{\\text{iter}}$}\n   \\STATE $m_n = \\alpha_n m_{n-1} + (1-\\alpha_n)  \\nabla f(x_{n-1},\\xi_n)$\n   \\STATE $v_n = \\beta_n v_{n-1} + (1-\\beta_n) \\nabla f(x_{n-1},\\xi_n)^{\\odot 2}$\n   \\STATE $r_n = \\alpha_n r_{n-1} + (1-\\alpha_n)$\n   \\STATE $\\bar r_n = \\beta_n \\bar r_{n-1} + (1-\\beta_n)$\n  \\STATE $\\hat m_{n} = m_{n}\/r_n$\n   \\COMMENT{bias correction step}\n   \\STATE $\\hat v_{n} = v_{n}\/\\bar r_n$\n   \\COMMENT{bias correction step}\n   \\STATE $x_{n} = x_{n-1} - \\gamma_n \\hat m_{n} \/ (\\varepsilon+\\sqrt{\\hat v_{n}}) \\,.$\n   \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\nThe algorithm generates a sequence $z_n=(x_n,m_n,v_n)$ with initial point $z_0=(x_0,0,0)$,\nwhere $x_0\\in\\bR^d$. Apart from the fact that the hyperparameters $(\\gamma_n,\\alpha_n,\\beta_n)$\nnow depend on $n$, the main difference w.r.t Algorithm~\\ref{alg:adam} lies in the expression of the\ndebiasing step. As noted in Remark~\\ref{rem:debiasing}, the aim is to rescale $m_n$ (resp. $v_n$)\nin such a way that the rescaled version $\\hat m_n$ (resp. $\\hat v_n$) is a convex combination of\npast stochastic gradients (resp. squared gradients). While in the constant step case the rescaling\ncoefficient is $(1-\\alpha^n)^{-1}$ (resp. $(1-\\beta^n)^{-1}$), the decreasing step case requires dividing\n$m_n$ by the coefficient $r_n=1-\\prod_{i=1}^n\\alpha_i$ (resp. $v_n$ by $\\bar r_n=1-\\prod_{i=1}^n\\beta_i$),\nwhich keeps track of the previous weights:\n$$\n\\hat m_n = \\frac{m_n}{r_n} = \\frac{\\sum_{k=1}^{n} \\rho_{n,k} \\nabla f(x_{k-1},\\xi_{k})}{\\sum_{k=1}^{n} \\rho_{n,k}}\\,,\n$$\nwhere for every $n,k$, $\\rho_{n,k} = \\alpha_n\\cdots\\alpha_{k+1}(1-\\alpha_k)$. A similar equation holds for $\\hat v_n$.\n\n\\subsection{Almost sure convergence\n  \\begin{assumption}[Stepsizes]\n    \\label{hyp:stepsizes}\n \n  The following holds.\n  \\begin{enumerate}[{\\sl i)}]\n  \\item For all $n \\in \\bN$, $\\gamma_n > 0$ and $\\gamma_{n+1}\/\\gamma_n\\to 1$,\n  \\item $\\sum_n \\gamma_n = +\\infty$ and $\\sum_n \\gamma_n^2 <+\\infty$,\n  \\item For all $n \\in \\bN$, $0 \\leq \\alpha_n \\leq 1$ and $0 \\leq \\beta_n \\leq 1$,\n  \\item There exist $a,b$ s.t. $0<b<4a$,\\,\n  $\\gamma_n^{-1}(1-\\alpha_n) \\to a$ and $\\gamma_n^{-1}(1-\\beta_n) \\to b$\\,.\n\n  \n  \n  \n \n  \\end{enumerate}\n  \\end{assumption}\n\n\n\n\n\n\n\n\n \\begin{theorem}\n \\label{thm:as_conv_under_stab}\nLet Assumptions~\\ref{hyp:model} to \\ref{hyp:S>0}, \\ref{hyp:iid} and~\\ref{hyp:stepsizes} hold.\nLet Assumption \\ref{hyp:moment-f}~\\ref{momentegal} hold with $p=4$.\nAssume that $F(\\cS)$ has an empty interior\nand that the random sequence $((x_n,m_n,v_n):n\\in \\bN)$\ngiven by Algorithm~\\ref{alg:adam-decreasing}\nis bounded, with probability one.\nThen, w.p.1, $\\lim_{n\\to\\infty}\n\\sd (x_n,\\cS)=0$, $lim_{n\\to\\infty}m_n = 0$\nand $lim_{n\\to\\infty} (S(x_n)-v_n)=0$.\nIf moreover $\\cS$ is finite or countable, then w.p.1, there exists $x^*\\in \\cS$\ns.t. $\\lim_{n\\to\\infty} (x_n,m_n,v_n) = (x^*,0,S(x^*))$.\n\n\n\n\n\n\n\n\n \\end{theorem}\nTh.~\\ref{thm:as_conv_under_stab} establishes the almost sure convergence of\n$x_n$ to the set of critical points of~$F$, under the assumption that the sequence\n$((x_n,m_n,v_n))$ is a.s. bounded. The next result provides a sufficient condition\nunder which almost sure boundedness holds.\n\\begin{assumption}\n\\label{hyp:stab}\n  The following holds.\n  \\begin{enumerate}[{\\sl i)}]\n\\item \\label{lipschitz} $\\nabla F$ is Lipschitz continuous.\n\\item \\label{momentgrowth} There exists $C>0$ s.t. for all $x \\in \\bR^d$, $\\bE[\\|\\nabla f(x,\\xi)\\|^2] \\leq C (1+F(x))$.\n\\item We assume the condition:\n\n\\lim\\sup_{n\\to\\infty}\n\\left(\\frac 1{\\gamma_n}-\\left(\\frac{1-\\alpha_{n+2}}{1-\\alpha_{n+1}}\\right)\\frac 1{\\gamma_{n+1}}\\right)\n< 2(a-\\frac b4)\\,,\n$\\\\%$$\nwhich is satisfied for instance if $b<4a$ and $1-\\alpha_{n+1} = a\\gamma_n$.\n  \\end{enumerate}\n\\end{assumption}\n \\begin{theorem}\n \\label{thm:stab}\n\nLet Assumptions~\\ref{hyp:model}, \\ref{hyp:coercive}, \\ref{hyp:iid}, \\ref{hyp:stepsizes} and \\ref{hyp:stab} hold.\nLet Assumption~\\ref{hyp:moment-f}~\\ref{momentegal} hold with $p=4$.\nThen, the sequence\n$((x_n,m_n,v_n):n\\in \\bN)$ given by Algorithm~\\ref{alg:adam-decreasing} is bounded with probability one.\n \\end{theorem}\n\n\\subsection{Central Limit Theorem}\n\n \\begin{assumption}\n   \\label{hyp:mean_field_tcl}\nLet $x^*\\in \\cS$. There exists a neighborhood $\\cV$ of $x^*$ s.t.\n\\begin{enumerate}[{\\sl i)}]\n\\item $F$ is twice continuously differentiable on $\\cV$,\n  and the Hessian $\\nabla^2 F(x^*)$ of $F$ at $x^*$ is positive\n  definite.\n\\item $S$ is continuously differentiable on $\\cV$.\n\\end{enumerate}\n \\end{assumption}\nDefine $\nD \\eqdef \\textrm{diag}\\left((\\varepsilon + \\sqrt{S_1(x^*)})^{-1},\\cdots,(\\varepsilon + \\sqrt{S_d(x^*)})^{-1}\\right)\\,.\n$\nLet $P$ be an orthogonal matrix s.t. the following spectral decomposition holds:\n$$\nD^{1\/2}\\nabla^2F(x^*)D^{1\/2} = P\\textrm{diag}(\\lambda_1,\\cdots,\\lambda_d)P^{-1}\\,,\n$$\nwhere $\\lambda_1, \\cdots,\\lambda_d$ are the (positive) eigenvalues of\n$D^{1\/2} \\nabla^2F(x^*)D^{1\/2}$.\nDefine\n\\begin{equation}\nH \\eqdef\n\\begin{pmatrix}\n  0 & -D & 0\\\\ a\\nabla^2F(x^*) & -a I_d & 0 \\\\ b \\nabla S(x^*) & 0 & -b I_d\n\\end{pmatrix}\\label{eq:H}\n\\end{equation}\nwhere $I_d$ represents the $d\\times d$ identity matrix and $\\nabla S(x^*)$ is the\nJacobian matrix of~$S$ at $x^*$.\nThe largest real part of the eigenvalues of $H$ coincides with $-L$, where\n\\begin{equation}\nL\\eqdef b\\wedge \\frac a2\\left( 1-\\sqrt{\\left(1-\\frac{4\\lambda_1}a\\right)\\vee 0}\\right) >0\\,.\\label{eq:L}\n\\end{equation}\nFinally, define the $3d\\times 3d$ matri\n\\begin{equation}\n  \\resizebox{.9\\hsize}{!}{$\nQ \\eqdef\n\\left( \\begin{array}[h]{cc}\n   0 &\n   \\begin{array}[h]{cc}\n     0 & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0\n   \\end{array}\\\\\n   \\begin{array}[h]{c}\n     0 \\\\ 0\n   \\end{array}\n & \\bE\\left[\n  \\begin{pmatrix}\n    a\\nabla f(x^*,\\xi) \\\\ b(\\nabla f(x^*,\\xi)^{\\odot 2}-S(x^*))\n  \\end{pmatrix}\\begin{pmatrix}\n    a\\nabla f(x^*,\\xi) \\\\ b(\\nabla f(x^*,\\xi)^{\\odot 2}-S(x^*))\n  \\end{pmatrix}^T\\right]\n \\end{array}\\right)\\,.\\label{eq:Q}\n $}\n\\end{equation}\n\\begin{assumption}\n\\label{hyp:step-tcl}\nThe following holds.\n\\begin{enumerate}[{\\sl i)}]\n\\item \\label{step-tcl-i} There exist $\\kappa \\in (0,1]$, $\\gamma_0>0$, s.t. the sequence $(\\gamma_n)$\nsatisfies\n$\\gamma_n = {\\gamma_0}\/{(n+1)^\\kappa}$ for all $n$.\nIf $\\kappa = 1$, we assume moreover that $\\gamma_0 > \\frac{1}{2L}$.\n\\item The sequences $\\left(\\frac{1}{\\gamma_n}(\\frac{1-\\alpha_n}{\\gamma_n}-a)\\right)$ and $\\left(\\frac{1}{\\gamma_n}(\\frac{1-\\beta_n}{\\gamma_n}-b)\\right)$ are bounded.\n\\end{enumerate}\n\\end{assumption}\n\n\\noindent For an arbitrary sequence $(X_n)$ of random variables on some Euclidean space, a probability measure\n$\\mu$ on that space and an event $\\Gamma$ s.t. $\\bP(\\Gamma)>0$, we say that $X_n$ converges in distribution\nto $\\mu$ \\emph{given $\\Gamma$} if the measures $\\bP (X_n\\in \\cdot\\,|\\Gamma)$ converge weakly to $\\mu$.\n\\begin{theorem}\n \\label{thm:clt}\nLet Assumptions~\\ref{hyp:model}, \\ref{hyp:S>0}, \\ref{hyp:iid}, \\ref{hyp:mean_field_tcl} and \\ref{hyp:step-tcl} hold true.\nLet Assumption~\\ref{hyp:moment-f}~\\ref{momentreinforce} hold with $p=4$.\nConsider the iterates $z_n=(x_n,m_n,v_n)$ given by Algorithm~\\ref{alg:adam-decreasing}. Set $z^*=(x^*,0,S(x^*))$.\nSet $\\zeta \\eqdef 0$ if $0<\\kappa<1$ and $\\zeta \\eqdef \\frac{1}{2 \\gamma_0}$ if $\\kappa =1$.\nAssume $\\bP(z_n \\to z^*)>0$. Then, given the event $\\{z_n\\to z^*\\}$,\nthe rescaled vector $\\sqrt{\\gamma_n}^{-1}(z_n-z^*)$\nconverges in distribution to a zero mean Gaussian distribution on $\\bR^{3d}$ with a covariance matrix $\\Sigma$\nwhich is solution to the Lyapunov equation: $  \\left(H + \\zeta I_{3d} \\right) \\Sigma + \\Sigma \\left( H^T +  \\zeta I_{3d} \\right) = - Q$.\nIn particular, given $\\{z_n\\to z^*\\}$, the vector $\\sqrt{\\gamma_n}^{-1}(x_n-x^*)$\nconverges in distribution to a zero mean Gaussian distribution with a covariance matrix $\\Sigma_1$ given by:\n\\begin{equation}\n\\Sigma_1 = D^{1\/2} P\n\\left(\n\\frac{C_{k,\\ell}}{ (1 - \\frac{2\\zeta}{a})(\\lambda_k+\\lambda_\\ell-2\\zeta + \\frac 2a \\zeta^2) +\\frac 1{2(a-2\\zeta)}(\\lambda_k-\\lambda_\\ell)^2}\n\n\\right)_{k,\\ell=1\\dots d}\nP^{-1}D^{1\/2}\\label{eq:cov}\n\\end{equation}\nwhere $C\\eqdef P^{-1}D^{1\/2}\\bE\\left(\\nabla f(x^*,\\xi)\\nabla f(x^*,\\xi)^T\\right)D^{1\/2}P$.\n\\end{theorem}\n\nThe following remarks are useful.\n\\begin{itemize}[leftmargin=*]\n\\item The variable $v_n$ has an impact on the limiting covariance $\\Sigma_1$ through its limit $S(x^*)$ (used to define $D$),\nbut the fluctuations of $v_n$ and the parameter $b$ have no effect on $\\Sigma_1$.\nAs a matter of fact, $\\Sigma_1$ coincides with the limiting covariance matrix that would have been obtained by considering\niterates of the form\n\\begin{equation*}\n     \\begin{cases}\n       x_{n+1} &= x_n - \\gamma_{n+1} p_{n+1} \\\\\n       p_{n+1} &= p_n + a\\gamma_{n+1}(D\\nabla f(x_n,\\xi_{n+1})-p_n) \\,,\n     \\end{cases}\n\\end{equation*}\n\nwhich can be interpreted as a preconditioned version of the stochastic heavy ball algorithm~\\cite{gadat2018stochastic}.\nOf course, the above iterates are not implementable because the preconditioning matrix $D$ is unknown.\n\\item \nWhen $a$ is large, $\\Sigma_1$ is close to the matrix $\\Sigma_1^{(0)}$ obtained\nwhen letting $a \\to +\\infty$ in Eq.~(\\ref{eq:cov}).\nThe matrix $\\Sigma_1^{(0)}$ is the solution to the  Lyapunov equation\n$$\n(D \\nabla^2F(x^*) - \\zeta I_d)  \\Sigma_1^{(0)} + \\Sigma_1^{(0)} (\\nabla^2F(x^*) D - \\zeta I_d) = D \\bE\\left(\\nabla f(x^*,\\xi)\\nabla f(x^*,\\xi)^T\\right) D\\,.\n$$\nThe matrix $\\Sigma_1^{(0)}$ can be interpreted as the asymptotic covariance matrix of the $x$-variable\nin the absence of the inertial term (that is, when one considers \\textsc{RmsProp} instead of \\adam).\nThe matrix $\\Sigma_1^{(0)}$ approximates $\\Sigma_1$ in the sense that $\\Sigma_1 = \\Sigma_1^{(0)}+\\frac 1a\\Delta + O(\\frac 1{a^2})$\nfor some symmetric matrix $\\Delta$ which can be explicited. The matrix $\\Delta$ is neither positive nor negative definite\nin general.\nThis suggests that the question of the potential benefit of the presence of an inertial term\nis in general problem dependent.\n\\item In the statement of Th.~\\ref{thm:clt},\nthe conditioning event $\\{z_n\\to z^*\\}$ can be replaced by the event $\\{x_n\\to x^*\\}$\nunder the additional assumption that $\\sum_n \\gamma_n^2 < +\\infty$.\n\\end{itemize}\n\n \\section{Related Works}\n \\label{sec:related-works}\n\n Although the idea of adapting the\n (per-coordinate) learning rates as a function of past gradient values\n is not new (see \\emph{e.g.} variable metric methods such as the BFGS\n algorithms),\n \\textsc{AdaGrad} \\cite{duchi2011adaptive} led the way to a new class of algorithms\n that are sometimes referred to as adaptive gradient methods. \\textsc{AdaGrad}\n consists of dividing the learning rate by the square root of the sum\n of previous gradients squared\n componentwise. \n The idea was to give larger learning rates to highly informative but\n infrequent features instead of using a fixed predetermined schedule.\n However, in practice, the division by the cumulative sum of squared\n gradients may generate small learning rates, thus freezing the\n iterates too early.  Several works proposed\n heuristical ways to set the learning rates using a less aggressive\n policy.\n The work \\cite{tieleman2012lecture} introduced an unpublished, yet popular, algorithm\n referred to as \\textsc{RmsProp} where the cumulative sum\n used in \\textsc{AdaGrad} is replaced by a moving average of squared\n gradients.\n\n\n\n \\adam\\ combines the advantages of both \\textsc{AdaGrad},\n \\textsc{RmsProp} and inertial methods\n\n As opposed to \\textsc{AdaGrad}, for which theoretical convergence guarantees exist\n \\cite{duchi2011adaptive,chen2018convergence,zhou2018convergence,ward2018adagrad},\n \\adam\\ is comparatively less studied.\n The initial paper \\cite{kingma2014adam} suggests a $\\mathcal{O}(\\frac{1}{\\sqrt{T}})$ average regret bound in the convex setting,\n but \\cite{j.2018on} exhibits a counterexample in contradiction with this statement.\n The latter counterexample implies that the average regret bound of \\adam\\ does\n not converge to zero. A first way to overcome the problem is to modify the \\adam\\ iterations\n themselves in order to obtain a vanishing average regret. This led \\cite{j.2018on}\n to propose a variant called \\textsc{AmsGrad} with the aim to recover, at least in the convex case, the sought guarantees.\n\n\n\n\n\n\n\n\n\n The work \\cite{balles2018dissecting} interprets \\adam\\ as a variance-adapted sign descent combining an update direction given by the sign and\n a magnitude controlled by a variance adaptation principle. A ``noiseless'' version\n of \\adam\\ is considered in \\cite{basu2018convergence}. Under quite specific values of the \\adam-hyperparameters, it is shown that for every $\\delta>0$,\n there exists some time instant\n for which the norm of the gradient of the objective\n at the current iterate is no larger than~$\\delta$.\n\n\n The recent paper \\cite{chen2018convergence} provides a similar result\n for \\textsc{AmsGrad} and \\textsc{AdaGrad}, but the generalization to \\adam\\ is subject\n to conditions which are not easily verifiable.\n The paper \\cite{zaheer2018adaptive} provides a convergence result for \\textsc{RmsProp}\n using the objective function $F$ as a Lyapunov function.\n\n However, our work suggests that unlike \\textsc{RmsProp},\n \\adam\\ does not admit $F$ as a Lyapunov function.\n This makes the approach of\n\n \\cite{zaheer2018adaptive} hardly generalizable to \\adam.\n Moreover, \\cite{zaheer2018adaptive} considers biased gradient estimates instead of the debiased\n estimates used in \\adam.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n In the present work, we study the behavior of an ODE, interpreted as the\n limit in probability of the (interpolated) \\adam\\ iterates as the stepsize tends to zero.\n Closely related continuous-time dynamical systems are also studied in \\cite{attouch2000heavy,cabot2009long}.\n We leverage the idea of approximating a discrete time stochastic system by a deterministic continuous one,\n often referred to as the ODE method.\n A recent work \\cite{gadat2018stochastic} fruitfully exploits this method to study\n a stochastic version of the celebrated heavy ball algorithm.\n We refer to \\cite{davis2018stochastic} for the reader interested in the non-differentiable setting\n with an analysis of the stochastic subgradient algorithm for non-smooth non-convex objective functions.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n Concomitant to the present paper, Da Silva and Gazeau \\cite{da2018general}\n (posted only four weeks after the first version of the present work)\n\n study the asymptotic behavior of a similar dynamical system as the one introduced here.\n They establish several results in continuous time, such as avoidance of traps\n as well as convergence rates in the convex case; such aspects are out of the scope of this paper.\n However, the question of the convergence of the (discrete-time) iterates is left open.\n In the current paper, we also exhibit a Lyapunov function which allows, amongst others, to draw useful conclusions on the effect of the\n debiasing step of \\adam\\,. Finally, \\cite{da2018general} studies a slightly modified version of \\adam\\, allowing to recover an\n ODE with a locally Lipschitz continuous vector field, whereas the original \\adam\\ algorithm \\cite{kingma2014adam} leads\n to an ODE with an irregular vector field. This technical issue is tackled in the present paper.\n\n\\section{Proofs of Section~\\ref{sec:continuous_time}}\n\\label{sec:proofs_cont_time}\n\n\\subsection{Preliminaries}\n\\label{subsec:setting}\nThe results in this section are not specific to the case where $F$ and $S$ are defined as in\nEq.~(\\ref{eq:F_and_S}): they are stated for\n\\emph{any} mappings $F$, $S$ satisfying the following hypotheses.\n\\begin{assumption}\n  \\label{hyp:F}\nThe function $F:\\bR^d\\to\\bR$ is s.t.: $F$ is continuously differentiable and\n$\\nabla F$ is locally Lipschitz continuous.\n\\end{assumption}\n\\begin{assumption}\n\\label{hyp:S}\nThe map $S:\\bR^d\\to [0,+\\infty)^d$ is locally Lipschitz continuous.\n\\end{assumption}\nIn the sequel, we consider the following generalization of Eq. (\\ref{eq:ode}) for any $\\eta >0$:\n\\begin{equation}\n  \\begin{array}[h]{l}\n\\dot z(t) = h(t+\\eta, z(t))\\,.\n\\end{array}\n\\tag{ODE\\mbox{$_\\eta$}}\n\\label{eq:odeeta}\\end{equation}\nWhen $\\eta=0$, Eq. (\\ref{eq:odeeta}) boils down to the equation of interest (\\ref{eq:ode}).\nThe choice $\\eta\\in (0,+\\infty)$ will be revealed useful to prove Th.~\\ref{th:exist-unique}.\nIndeed, for $\\eta>0$, a solution to Eq. (\\ref{eq:odeeta}) can be shown to exist (on some interval) due to the continuity of\nthe map $h(\\,.+\\eta,\\,.\\,)$. Considering a family of such solutions indexed by $\\eta\\in (0,1]$,\nthe idea is to prove the existence of a solution to (\\ref{eq:ode}) as a cluster point of the latter family when $\\eta\\downarrow 0$.\nIndeed, as the family is shown to be equicontinuous, such a cluster point does exist thanks to the Arzel\u00e0-Ascoli theorem.\nWhen $\\eta=+\\infty$,\nEq. (\\ref{eq:odeeta}) rewrites\n\\begin{equation}\n  \\label{eq:ode-a}\n  \\begin{array}[h]{l}\n\\dot z(t) = h_\\infty(z(t))\\,,\n\\end{array}\n\\tag{ODE\\mbox{$_\\infty$}}\n\\end{equation}\nwhere $h_\\infty(z)\\eqdef \\lim_{t\\to \\infty} h(t,z)$.\nIt is useful to note that for $(x,m,v)\\in \\cZ_+$,\n\\begin{equation}\n  \\label{eq:h_infty}\nh_{\\infty}((x,m,v)) = \\left(-m \/ (\\varepsilon+\\sqrt{v})\\,,\\, a (\\nabla F(x)-m) \\,,\\,b (S(x)-v) \\right)\\,.\n\\end{equation}\nContrary to Eq. (\\ref{eq:ode}), Eq.~(\\ref{eq:ode-a}) defines an  autonomous ODE.\nThe latter admits a unique global solution for any initial condition in $\\cZ_+$,\nand defines a dynamical system $\\cD$. We shall exhibit a strict Lyapunov function\nfor this dynamical system $\\cD$, and deduce that any solution to (\\ref{eq:ode-a}) converges\nto the set of equilibria of $\\cD$ as $t\\to\\infty$.\nOn the otherhand, we will prove that the solution to (\\ref{eq:ode}) with a proper initial condition is a so-called asymptotic pseudotrajectory (APT) of $\\cD$. Due to the\nexistence of a strict Lyapunov function, the APT shall inherit the convergence behavior of the autonomous system as $t\\to\\infty$,\nwhich will prove Th.~\\ref{th:cv-adam}.\n\nIt is convenient to extend the map $h:(0,+\\infty)\\times \\cZ_+\\to\\cZ$ on  $(0,+\\infty)\\times \\cZ\\to\\cZ$ by\nsetting $h(t,(x,m,v))\\eqdef h(t,(x,m,|v|))$ for every $t>0$, $(x,m,v)\\in \\cZ$.\nSimilarly, we extend $h_\\infty$ as $h_\\infty((x,m,v)) \\eqdef h_\\infty((x,m,|v|))$.\nFor any $T\\in (0,+\\infty]$ and any $\\eta\\in [0,+\\infty]$, we say that a map $z:[0,T)\\to \\cZ$ is a solution\nto (\\ref{eq:odeeta}) on $[0,T)$ with initial condition $z_0\\in \\cZ_+$,\nif $z$ is continuous on $[0,T)$, continuously differentiable\non $(0,T)$, and if (\\ref{eq:odeeta}) holds for all $t\\in (0,T)$.\nWhen $T=+\\infty$, we say that the solution is global.\nWe denote by $Z^\\eta_T(z_0)$ the subset of $C([0,T),\\cZ)$ formed by the solutions to (\\ref{eq:odeeta})\non $[0,T)$ with initial condition $z_0$.\nFor any $K\\subset\\cZ_+$, we define $Z^\\eta_T(K)\\eqdef \\bigcup_{z\\in K}Z^\\eta_T(z)$.\n\n\n\\begin{lemma}\n  \\label{lem:m-v-derivables-en-zero}\n  Let Assumptions~\\ref{hyp:F} and \\ref{hyp:S} hold.  Consider $x_0\\in \\bR^d$,\n  $T\\in (0,+\\infty]$ and let $z\\in Z_T^0((x_0,0,0))$,\n  which we write $z(t) = (x(t),m(t),v(t))$.  Then, $z$\n  is continuously differentiable on $[0,T)$,\n  $\\dot m(0)=a\\nabla F(x_0)$, $\\dot v(0)=bS(x_0)$ and\n$\n\\dot x(0) =  \\frac{-\\nabla F(x_0)}{\\varepsilon + \\sqrt{S(x_0)}}.\n$\n\\end{lemma}\n\\begin{proof}\nBy definition of $z(\\,.\\,)$, $m(t)=\\int_0^ta(\\nabla F(x(s))-m(s))ds$ for all $t\\in [0,T)$\n(and a similar relation holds for $v(t)$).\nThe integrand being continuous, it holds\nthat $m$ and $v$ are differentiable at  zero and $\\dot m(0)=a\\nabla F(x_0)$, $\\dot v(0)=bS(x_0)$.\nSimilarly, $x(t) = x_0+\\int_0^t h_x(s,z(s))ds$, where\n$h_x(s,z(s)) \\eqdef -(1-e^{-as})^{-1}m(s)\/(\\varepsilon+\\sqrt{(1-e^{-bs})^{-1}v(s)})\\,.$\nNote that $m(s)\/s \\to \\dot m(0) = a\\nabla F(x_0) $ as $s\\downarrow 0$.\nThus, $(1-e^{-as})^{-1}m(s)\\to \\nabla F(x_0)$ as $s\\to 0$. Similarly,\n$(1-e^{-bs})^{-1}v(s)\\to S(x_0)$. It follows that\n$h_x(s,z(s))\\to -(\\varepsilon+\\sqrt{S(x_0)})^{-1}\\nabla F(x_0)$.\nThus, $s\\mapsto h_x(s,z(s))$\ncan be extended\nto a continuous map on $[0,T)\\to\\bR^d$ and the differentiability of $x$ at zero\nfollows.\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{lem:v-positif}\nLet Assumptions~\\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\nFor every $\\eta\\in [0,+\\infty]$, $T\\in (0,+\\infty]$,  $z_0\\in \\cZ_+$,  $z\\in Z_T^\\eta(z_0)$,\nit holds that $z((0,T))\\subset \\cZ_+^*$.\n\\end{lemma}\n\\begin{proof}\nSet $z(t) = (x(t),m(t),v(t))$ for all $t$. Consider $i\\in \\{1,\\dots,d\\}$.\nAssume by contradiction that there exists $t_0\\in (0,T)$ s.t.\n$v_i(t_0)<0$. Set $\\tau\\eqdef\\sup\\{t\\in [0,t_0]:v_i(t)\\geq 0\\}$.\nClearly, $\\tau<t_0$ and $v_i(\\tau)=0$ by the continuity of $v_i$.\nSince $v_i(t)\\leq 0$ for all $t\\in (\\tau,t_0]$, it holds that $\\dot v_i(t) = b(S_i(x(t))-v_i(t))$\nis nonnegative for all  $t\\in (\\tau,t_0]$. This contradicts the fact that $v_i(\\tau)>v_i(t_0)$.\nThus, $v_i(t)\\geq 0$ for all $t\\in [0,T)$.\nNow assume by contradiction that there exists $t\\in (0,T)$  s.t.\n$v_i(t)=0$. Then, $\\dot v_i(t)=bS_i(x(t))>0$.\nThus,\n$\n\\lim_{\\delta\\downarrow 0} \\frac{v_i(t-\\delta)}{-\\delta} = bS_i(x(t))\\,.\n$\nIn particular, there exists $\\delta>0$ s.t.\n$v_i(t-\\delta)  \\leq  -\\frac{\\delta b}2S_i(x(t))\\,.$ This contradicts the first point.\n\\end{proof}\n\n Recall the definitions of $V$ and $U$ from Eqs.~(\\ref{eq:V}) and (\\ref{eq:U}).\n Clearly, $U_\\infty(v)\\eqdef\\lim_{t\\to \\infty} U(t,v)=a(\\varepsilon+\\sqrt{v})$ is well defined for every $v\\in [0,+\\infty)^d$.\n Hence, we can also define $V_\\infty(z)\\eqdef \\lim_{t\\to \\infty} V(t,z)$ for every $z\\in \\cZ_+$.\n\n\n\n\n\\begin{lemma}\n\\label{lem:V}\n \n  Let Assumptions~\\ref{hyp:F} and \\ref{hyp:S} hold.\n   Assume that $0< b\\leq 4a$.\nConsider $(t,z)\\in (0,+\\infty)\\times \\cZ_+^*$ and set $z=(x,m,v)$.\nThen, $V$ and $V_\\infty$ are  differentiable at points  $(t,z)$ and $z$ respectively. Moreover,\n$\\ps{\\nabla V_\\infty(z),h_{\\infty}(z)}\\leq -\\varepsilon \\left\\|\\frac {am}{U_\\infty(v)}\\right\\|^2\\,$ and\n\\begin{equation*}\n\\ps{\\nabla V(t,z), (1,h(t,z))} \\leq -\\frac{\\varepsilon }2\\left\\|\\frac{a\\,m}{U(t,v)}\\right\\|^2\\,.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nWe only prove the second point, the proof of the first point follows the same line.\n Consider $(t,z)\\in (0,+\\infty)\\times \\cZ_+^*$.\nWe decompose\n$\\ps{\\nabla V(t,z), (1,h(t,z))} = \\partial_tV(t,z)\n+\\ps{\\nabla_z V(t,z),h(t,z)}$. After tedious but straightforward derivations, we get:\n\\begin{equation}\n  \\label{eq:partial-t}\n  \\resizebox{0.99\\hsize}{!}{$\n  \\partial_t V(t,z) =- \\sum_{i=1}^d \\frac{a^2m_i^2}{U(t,v_i)^2}\\left(\\frac{e^{-at}\\varepsilon}2+\n\\left(\\frac{e^{-at}}2-\\frac{be^{-bt}(1-e^{-at})}{4a(1-e^{-bt})}\\right)\\sqrt{\\frac{v_i}{1-e^{-bt}}}\\right)\\,,\n$}\n\\end{equation}\nwhere $U(t,v_i)=a(1-e^{-at})\\left(\\varepsilon+\\sqrt{\\frac{v_i}{1-e^{-bt}}}\\right)$ and $\\ps{\\nabla_z V(t,z),h(t,z)}$ is equal to:\n\\begin{equation*}\n    \\sum_{i=1}^d \\frac{-a^2m_i^2(1-e^{-at})}{U(t,v_i)^2}\n  \\left(\\varepsilon\n+ (1-\\frac b{4a})\\sqrt{\\frac{v_i}{1-e^{-bt}}}\n+\\frac{bS_i(x)}{4a\\sqrt{v_i(1-e^{-bt})}}\n\\right)\\,.\n\\end{equation*}\nUsing that $S_i(x)\\geq 0$, we obtain:\n\\begin{equation}\n\\ps{\\nabla V(t,z), (1,h(t,z))} \\leq  -\\sum_{i=1}^d \\frac{a^2m_i^2}{U(t,v_i)^2}\\left(\n(1-\\frac{e^{-at}}2)\\varepsilon+c_{a,b}(t)\\sqrt{\\frac{v_i}{1-e^{-bt}}}\n\\right)\\,,\\label{eq:ineg-V}\n\\end{equation}\nwhere $c_{a,b}(t)\\eqdef 1-\\frac{e^{-at}}2-\\frac b{4a}\\frac{1-e^{-at}}{1-e^{-bt}}\\,.$\nUsing inequality $1-{e^{-at}}\/2\\geq 1\/2$ in (\\ref{eq:ineg-V}), the inequality~(\\ref{eq:ineg-V})\nproves the Lemma, provided that one is able to show that $c_{a,b}(t)\\geq 0$, for all $t>0$\nand all $a,b$ satisfying $0< b\\leq 4a$. We prove this last statement.\nIt can be shown that the function $b\\mapsto c_{a,b}(t)$ is decreasing on $[0,+\\infty)$.\nHence, $c_{a,b}(t)\\geq c_{a,4a}(t)$. Now, $c_{a,4a}(t) = q(e^{-at})$ where $q:[0,1)\\to\\bR$ is the\nfunction defined for all $y\\in [0,1)$ by $q(y) = y \\left(y^4-2y^3+1\\right)\/(2(1-y^4))\\,$.\nHence $q \\geq 0$. Thus, $c_{a,b}(t)\\geq q(e^{-at})\\geq 0$.\n\\end{proof}\n\n\\subsection{Proof of Th.~\\ref{th:exist-unique}}\n\n\\subsubsection{Boundedness}\nDefine $\\cZ_0 \\eqdef \\{(x,0,0):x\\in \\bR^d\\}$.\nLet $\\bar e:(0,+\\infty)\\times \\cZ_+\\to\\cZ_+$ be defined by\n$\\bar e(t,z)\\eqdef (x,m\/(1-e^{-at}),v\/(1-e^{-bt}))\\,$\nfor every $t>0$ and every $z=(x,m,v)$ in $\\cZ_+$.\n\n\\begin{proposition}\n\\label{prop:adam-bounded}\nLet Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\nAssume that $0< b\\leq 4a$.\nFor every $z_0\\in \\cZ_0$, there exists a compact set $K\\subset \\cZ_+$ s.t.\nfor all $\\eta\\in [0,+\\infty)$, all $T\\in (0,+\\infty]$ and all $z\\in Z_T^\\eta(z_0)$,\n$\\left\\{\\bar e(t+\\eta,z(t)) :t\\in (0,T)\\right\\} \\subset K\\,.$\nMoreover, choosing $z_0$ of the form $z_0=(x_0,0,0)$ and $z(t) = (x(t), m(t),v(t))$, it holds that $F(x(t))\\leq F(x_0)$ for all $t\\in [0,T)$.\n\\end{proposition}\n\n\\begin{proof}\nLet $\\eta\\in [0,+\\infty)$.\nConsider a solution $z_\\eta(t) = (x_\\eta(t),m_\\eta(t),v_\\eta(t))$  as in the statement, defined on some interval $[0,T)$.\nDefine\n$\\hat m_\\eta(t) = m_\\eta(t)\/(1-e^{-a(t+\\eta)})$,\n$\\hat v_\\eta(t) = v_\\eta(t)\/(1-e^{-b(t+\\eta)})$.\nBy Lemma~\\ref{lem:v-positif}, $t\\mapsto V(t+\\eta,z(t))$ is continuous on $[0,T)$, and\ncontinuously differentiable on $(0,T)$.\nBy Lemma~\\ref{lem:V}, $\\dot V(t+\\eta,z_\\eta(t)) \\leq 0$ for all $t>0$.\nAs a consequence, $t\\mapsto V(t+\\eta,z_\\eta(t))$ is non-increasing on $[0,T)$.\nThus, for all $t\\geq 0$, $F(x_{\\eta}(t))\\leq \\lim_{t'\\downarrow 0}V(t'+\\eta,z_\\eta(t'))$. Note that\n$\n  V(t+\\eta,z_\\eta(t)) \\leq F(x_\\eta(t))+\\frac 12 \\sum_{i=1}^d\n  \\frac{m_{\\eta,i}(t)^2}{a(1-e^{-a(t+\\eta)})\\varepsilon}\\,. \\label{eq:majV}\n$\nIf $\\eta>0$, every term in the sum in the righthand side\ntends to zero, upon noting that\n$m_{\\eta}(t)\\to 0$ as $t\\to 0$.\nThe statement still holds if $\\eta=0$. Indeed, by Lemma~\\ref{lem:m-v-derivables-en-zero},\nfor a given $i\\in \\{1,\\dots,d\\}$, there exists $\\delta>0$ s.t. for all $0<t<\\delta$,\n$m_{\\eta,i}(t)^2\\leq 2 a^2 (\\partial_i F(x_0))^2 t^2$\nand $1-e^{-at}\\geq (at)\/2$. As a consequence, each term of the sum\nis no larger than $4 (\\partial_i F(x_0))^2 t\/\\varepsilon$, which tends to zero as $t\\to 0$.\nWe conclude that for all $t\\geq 0$, $F(x_\\eta(t))\\leq F(x_0)$.\nIn particular, $\\{x_\\eta(t):t\\in [0,T)\\}\\subset \\{F\\leq F(x_0)\\}$, the latter set being bounded\nby Assumption~\\ref{hyp:coercive}.\n\n\nWe prove that $v_{i,\\eta}(t)$ is (upper)bounded.\nDefine $R_{i} \\eqdef \\sup  S_i(\\{F\\leq F(x_0)\\})$, which is finite by continuity of $S$.\nAssume by contradiction that the set $\\{t\\in [0,T):v_{\\eta,i}(t)\\geq R_{i}+1\\}$ is non-empty, and denote\nits infimum by $\\tau$. By continuity of $v_{\\eta,i}$, one has\n$v_{\\eta,i}(\\tau) =  R_{i}+1$. This by the way implies that $\\tau>0$. Hence,\n$\\dot v_{\\eta,i}(\\tau) = b(S_i(x_\\eta(\\tau))-v_{\\eta,i}(\\tau)) \\leq -b$.\nThis means that there exists $\\tau'<\\tau$ s.t. $v_{\\eta,i}(\\tau')>v_{\\eta,i}(\\tau)$, which contradicts the definition of $\\tau$.\nWe have shown that $v_{\\eta,i}(t)\\leq R_i+1$ for all $t\\in (0,T)$.\nIn particular, when $t\\geq 1$, $\\hat v_{\\eta,i}(t) = v_{\\eta,i}(t)\/(1-e^{-bt}) \\leq (R_i+1)\/(1-e^{-b})\\,.$\nConsider $t\\in (0,1\\wedge T)$.\nBy the mean value theorem, there exists $\\tilde t_\\eta\\in [0,t]$ s.t. $v_{\\eta,i}(t) = \\dot v_{\\eta,i}(\\tilde t_\\eta)t$.\nThus, $v_{\\eta,i}(t) \\leq b S_i(x(\\tilde t_\\eta)) t\\leq b R_i t$. Using that the map $y\\mapsto y\/(1-e^{-y})$ is increasing on $(0,+\\infty)$,\nit holds that for all $t\\in (0,1\\wedge T)$,\n$\\hat v_{\\eta,i}(t)\n\\leq bR_i \/(1-e^{-b})\\,.$\nWe have shown that, for all $t\\in (0,T)$ and all $i\\in \\{1,\\dots,d\\}$, $0\\leq \\hat v_{\\eta,i}(t)\\leq M$, where\n$M\\eqdef (1-e^{-b})^{-1}(1+ b)(1+\\max\\{R_\\ell:\\ell\\in \\{1,\\dots,d\\})$.\n\nAs $V(t+\\eta,z_\\eta(t))\\leq F(x_0)$, we obtain: $F(x_0) \\geq F(x_\\eta(t))+\\frac 12\n\\left\\|m_\\eta(t)\\right\\|^2_{U(t+\\eta,v_\\eta(t))^{-1}}$.\nThus, $F(x_0) \\geq \\inf F+\\frac 1{2a(\\varepsilon+\\sqrt{M})} \\|m_{\\eta}(t)\\|^2\\,$.\nTherefore, $m_\\eta(\\,.\\,)$ is bounded on $[0,T)$, uniformly in $\\eta$.\nThe same holds for $\\hat m_\\eta$ by using the mean value theorem\nin the same way as for $\\hat v_\\eta$. The proof is complete.\n\\end{proof}\n\n\n\\begin{proposition}\n\\label{prop:bounded}\nLet Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\nAssume that $0< b\\leq 4a$.\nLet $K$ be a compact subset of $\\cZ_+$.\nThen, there exists an other compact set $K'\\subset \\cZ_+$ s.t.\nfor every $T\\in (0,+\\infty]$ and every $z\\in Z_{T}^\\infty(K)$,\n$z([0,T))\\subset K'$.\n\\end{proposition}\n\\begin{proof}\nThe proof follows the same line as Prop.~\\ref{prop:adam-bounded} and is omitted.\n\\end{proof}\n\n\n\nFor any $K\\subset \\cZ_+$, define $v_{\\min}(K)\\eqdef\\inf\\{v_k: (x,m,v)\\in K,i\\in \\{1,\\dots,d\\}\\}$.\n\\begin{lemma}\n\\label{lem:v-lowerbound}\nUnder Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S},\nthe following holds true\n\\begin{enumerate}[{\\it i)},leftmargin=*]\n\\item For every compact set $K\\subset \\cZ_+$, there exists $c>0$, s.t. for every $z\\in Z^\\infty_{\\infty}(K)$, of the form\n$z(t)= (x(t),m(t),v(t))$, $v_i(t)\\geq c \\min\\left(1 ,\\frac{v_{\\min}(K)}{2c}+ t\\right)\\qquad(\\forall t\\geq 0, \\forall i\\in\\{1,\\dots,d\\})\\,.$\n\\item For every $z_0\\in \\cZ_0$, there exists $c>0$ s.t. for every $\\eta\\in [0,+\\infty)$ and every $z\\in Z_\\infty^\\eta(z_0)$,\n$v_i(t)\\geq c\\min(1,t)\\qquad(\\forall t\\geq 0, \\forall i\\in\\{1,\\dots,d\\})\\,.$\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nWe prove the first point. Consider a compact set $K\\subset \\cZ_+$.\nBy Prop.~\\ref{prop:bounded}, one can find a compact set $K'\\subset \\cZ_+$ s.t.\nfor every $z\\in Z^\\infty_{\\infty}(K)$, it holds that $\\{z(t):t\\geq 0\\}\\subset K'$.\nDenote by $L_S$ the Lipschitz constant of $S$ on the compact set $\\{x:(x,m,v)\\in K'\\}$.\nIntroduce the constants $M_1\\eqdef \\sup\\{\\|m\/(\\varepsilon + \\sqrt v)\\|_\\infty:(x,m,v)\\in K'\\}$,\n$M_2\\eqdef \\sup\\{\\|S(x)\\|_\\infty:(x,m,v)\\in K'\\}$.\nThe constants $L_S, M_1, M_2$ are finite.\nNow consider a global solution $z(t)=(x(t),m(t),v(t))$ in $Z^\\infty_{\\infty}(K)$.\nChoose $i\\in \\{1,\\dots,d\\}$ and consider $t\\geq 0$. By the mean value theorem,\nthere exists $t'\\in [0,t]$ s.t. $v_i(t) = v_i(0)+\\dot v_i(t')t$. Thus,\n$ v_i(t) = v_i(0) + \\dot v_i(0) t + b(S_i(x(t')) - v_i(t') - S_i(x(0))+ v_i(0)) t$,\nwhich in turn implies\n$ v_i(t)\\geq v_i(0) + \\dot v_i(0) t - b L_S\\|x(t')-x(0)\\|t - b  |v_i(t') - v_i(0)| t$.\nUsing again the mean value theorem, for every $\\ell\\in \\{1,\\dots,d\\}$, there exists $t''\\in [0,t']$ s.t.\n$\n|x_\\ell(t')-x_\\ell(0)| = t' |\\dot x_\\ell(t'')| \\leq t M_1\\,.\n$\nTherefore, $\\|x(t')-x(0)\\|\\leq \\sqrt d M_1 t$. Similarly, there exists $\\tilde t$ s.t.:\n$|v_i(t') - v_i(0)|=  t'|\\dot v_i(\\tilde t)|\\leq t'b S_i(x(\\tilde t)) \\leq t bM_2\\,.$\nPutting together the above inequalities, $v_i(t) \\geq  v_i(0) (1-bt) + bS_i(x(0)) t - bC t^2 \\,$,\nwhere $C\\eqdef (M_2+L_S\\sqrt d M_1)$.\nFor every $t\\leq 1\/(2b)$, $v_i(t) \\geq \\frac{v_{\\min}}{2} + tbC\\left(\\frac{S_{\\min}}C  - t\\right) \\,,$\nwhere we defined $S_{\\min}\\eqdef\\inf\\{S_i(x):i\\in \\{1,\\dots,d\\}, (x,m,v)\\in K\\}$.\nSetting $\\tau \\eqdef 0.5\\min(1\/b,S_{\\min}\/C)$,\n\\begin{equation}\n\\forall t\\in [0,\\tau],\\  v_i(t) \\geq \\frac{v_{\\min}}{2} + \\frac{bS_{\\min}t}{2}\\,.\\label{eq:vlin}\n\\end{equation}\nSet $\\kappa_1\\eqdef 0.5(v_{\\min} + bS_{\\min}\\tau)$. Note that $v_i(\\tau)\\geq \\kappa_1$.\nDefine $S_{\\min}'\\eqdef\\inf\\{S_i(x):i\\in \\{1,\\dots,d\\}, (x,m,v)\\in K'\\}\\,.$\nNote that $S_{\\min}'>0$ by Assumptions~\\ref{hyp:S} and \\ref{hyp:S>0}.\nFinally, define $\\kappa = 0.5\\min(\\kappa_1,S_{\\min}')$.\nBy contradiction, assume that the set $\\{t\\geq \\tau : v_i(t)<\\kappa\\}$ is non-empty, and\ndenote by $\\tau'$ its infimum. It is clear that $\\tau'>\\tau$ and\n$v_i(\\tau')=\\kappa$. Thus, $b^{-1}\\dot v_i(\\tau') =S_i(x(\\tau'))-\\kappa$.\nWe obtain that $b^{-1}\\dot v_i(\\tau') \\geq 0.5S_{\\min}'>0$.\nAs a consequence, there exists $t\\in (\\tau,\\tau')$ s.t. $v_i(t)<v_i(\\tau')$. This contradicts\nthe definition of $\\tau'$. We have shown that for all $t\\geq \\tau$, $v_i(t)\\geq \\kappa$. Putting this together\nwith Eq.~(\\ref{eq:vlin}) and using that $\\kappa\\leq v_{\\min} + bS_{\\min}\\tau$,\nwe conclude that:\n$\\forall t\\geq 0,\\ v_i(t)\\geq \\min\\left(\\kappa\\,, \\frac{v_{\\min}}{2} + \\frac{bS_{\\min}t}{2}\\right)\\,.$\nSetting $c\\eqdef \\min(\\kappa,bS_{\\min}\/2)$, the result follows.\n\nWe prove the second point.  By Prop.~\\ref{prop:adam-bounded}, there exists a compact set $K\\subset \\cZ_+$ s.t.\nfor every $\\eta\\geq 0$, every $z\\in Z_\\infty^\\eta(x_0)$ of the form $z(t) = (x(t),m(t),v(t))$\nsatisfies\n$\\{(x(t),\\hat m(t),\\hat v(t)):t\\geq 0\\}\\subset K$,\nwhere $\\hat m(t) = m(t)\/(1-e^{-a(t+h)})$ and $\\hat v(t) = v(t)\/(1-e^{-b(t+h)})$.\nDenote by $L_S$ the Lipschitz constant of $S$ on the set $\\{x:(x,m,v)\\in K\\}$.\nIntroduce the constants $M_1\\eqdef \\sup\\{\\|m\/(\\varepsilon + \\sqrt v)\\|_\\infty:(x,m,v)\\in K\\}$,\n$M_2\\eqdef \\sup\\{\\|S(x)\\|_\\infty:(x,m,v)\\in K'\\}$. These constants being introduced,\nthe rest of the proof follows the same line as the proof of the first point.\n\\end{proof}\n\n\\subsubsection{Existence}\n\\label{sec:existence-adam}\n\n\\begin{corollary}\n\\label{coro:existence}\nLet Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\nAssume that $0< b\\leq 4a$.\nFor every $z_0\\in \\cZ_+$,$Z_{\\infty}^\\infty(z_0)\\neq\\emptyset$.\nFor every $(z_0,\\eta)\\in \\cZ_0\\times (0,+\\infty)$,$Z_{\\infty}^\\eta(z_0)\\neq\\emptyset$.\n\\end{corollary}\n\\begin{proof}\nWe prove the first point (the proof of the second point follows the same line).\nUnder Assumptions~\\ref{hyp:F} and \\ref{hyp:S}, $h_\\infty$ is continuous.\nTherefore, Cauchy-Peano's theorem guarantees the existence of a solution to the (\\ref{eq:ode}) issued from $z_0$,\nwhich we can extend over a maximal interval of existence $[0,T_{\\max})$\nWe conclude that the solution is global ($T_{\\max} = +\\infty$) using the boundedness of the solution given by Prop.~\\ref{prop:bounded}.\n\\end{proof}\n\\begin{lemma}\n\\label{lem:equicont-eta}\n \n  Let Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\n  Assume that $0< b\\leq 4a$.\nConsider $z_0\\in \\cZ_0$. Denote by $(z_\\eta:\\eta\\in (0,+\\infty))$ a family of functions on $[0,+\\infty)\\to \\cZ_+$\ns.t. for every $\\eta>0$, $z_\\eta\\in Z_\\infty^\\eta(z_0)$.\nThen,  $(z_\\eta)_{\\eta>0}$ is equicontinuous.\n\\end{lemma}\n\\begin{proof}\nFor every such solution $z_\\eta$, we set $z_\\eta(t)=(x_\\eta(t),m_\\eta(t),v_\\eta(t))$ for all $t\\geq 0$,\nand define $\\hat m_\\eta$ and $\\hat v_\\eta$ as in Prop.~\\ref{prop:adam-bounded}.\nBy Prop.~\\ref{prop:adam-bounded}, there exists a constant $M_1$ s.t. for all $\\eta>0$ and all $t\\geq 0$,\n$\\max(\\|x_\\eta(t)\\|,\\|\\hat m_\\eta(t)\\|_\\infty,\\|\\hat v_\\eta(t)\\|)\\leq M_1$.\nUsing the continuity of $\\nabla F$ and $S$, there exists an other finite constant $M_2$ s.t.\n$M_2\\geq \\sup\\{\\|\\nabla F(x)\\|_\\infty:x\\in \\bR^d, \\|x\\|\\leq M_1\\}$ and\n$M_2\\geq \\sup\\{\\|S(x)\\|_\\infty:x\\in \\bR^d, \\|x\\|\\leq M_1\\}$.\nFor every $(s,t)\\in [0,+\\infty)^2$,  we have for all $i\\in \\{1,\\dots,d\\}$,\n$|x_{\\eta,i}(t)-x_{\\eta,i}(s)|\\leq \\int_s^t\\left|\\frac{\\hat m_{\\eta,i}(u)}{\\varepsilon + \\sqrt{\\hat v_{\\eta,i}(u)}}\\right|du\\,\\leq \\frac{M_1}\\varepsilon |t-s|$,\nand similarly $|m_{\\eta,i}(t)-m_{\\eta,i}(s)| \\leq a(M_1+M_2)|t-s|$,\n$|v_{\\eta,i}(t)-v_{\\eta,i}(s)|\\leq b(M_1+M_2)|t-s|$\\,.\nTherefore, there exists a constant $M_3$, independent from $\\eta$, s.t. for all $\\eta>0$ and all $(s,t)\\in [0,+\\infty)^2$,\n$\\|z_\\eta(t)-z_\\eta(s)\\|\\leq M_3 |t-s|$\n\\end{proof}\n\n\n\\begin{proposition}\n  \\label{prop:existence-adam}\n \n  Let Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\n  Assume that $0< b\\leq 4a$.\nFor every  $z_0\\in \\cZ_0$, $Z_\\infty^0(z_0)\\neq \\emptyset$ \\emph{i.e.},\n(\\ref{eq:ode}) admits a global solution issued from $z_0$.\n\\end{proposition}\n\\begin{proof}\n  By Cor.~\\ref{coro:existence}, there exists  a family $(z_\\eta)_{\\eta>0}$ of functions on $[0,+\\infty)\\to \\cZ$\ns.t. for every $\\eta>0$, $z_\\eta\\in Z^\\eta_\\infty(z_0)$.\nWe set as usual $z_\\eta(t)=(x_\\eta(t),m_\\eta(t),v_\\eta(t))$. By Lemma~\\ref{lem:equicont-eta},\nand the Arzel\u00e0-Ascoli theorem, there exists a map $z:[0,+\\infty)\\to \\cZ$ and a sequence $\\eta_n\\downarrow 0$ s.t.\n$z_{\\eta_n}$ converges to $z$ uniformly on compact sets, as $n\\to\\infty$. Considering some fixed scalars $t>s> 0$,\n$z(t) = z(s) + \\lim_{n\\to\\infty}\\int_s^t h(u+\\eta_n, z_{\\eta_n}(u))du\\,.$\nBy Prop.~\\ref{prop:adam-bounded}, there exists a compact set $K\\subset \\cZ_+$ s.t.\n$\\{z_{\\eta_n}(t):t\\geq 0\\}\\subset K$ for all $n$.\nMoreover, by Lemma~\\ref{lem:v-lowerbound}, there exists a constant $c>0$ s.t.\nfor all $n$ and all $u\\geq 0$, $v_{\\eta_n,k}(u)\\geq c \\min(1,u)$.\nDenote by $\\bar K\\eqdef K\\cap (\\bR^d\\times\\bR^d\\times [c\\min(1,s),+\\infty)^d)$.\nIt is clear that $\\bar K$ is a compact subset of $\\cZ_+^*$.\nSince $h$ is continuously differentiable on the set\n$[s,t]\\times \\bar K$, it is Lipschitz continuous on that set. Denote by $L_h$ the corresponding\nLipschitz constant. We obtain:\n$$\n\\int_s^t\\|h(u+\\eta_n, z_{\\eta_n}(u)) - h(u, z(u))\\|du \\leq L_h\\left(\\eta_n + \\sup_{u\\in [s,t]}\\|z_{\\eta_n}(u)-z(u)\\|\\right)(t-s)\\,,\n$$\nand the righthand side converges to zero. As a consequence, for all $t>s$,\n$\nz(t) = z(s) + \\int_s^t h(u, z(u))du\\,.\n$ Moreover, $z(0)=z_0$. This proves that $z\\in Z^0_\\infty(z_0)$.\n\\end{proof}\n\n\\subsubsection{Uniqueness}\n\n\\begin{proposition}\n  \\label{prop:unique-adam}\nLet Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\nAssume  $b\\leq 4a$.\nFor every $z_0\\in\\cZ_0$, $Z_\\infty^0(z_0)$ is a singleton\n\\emph{i.e.}, there exists a unique global solution to (\\ref{eq:ode})\nwith initial condition $z_0$.\n\\end{proposition}\n\\begin{proof}\n \n  Consider solutions $z$ and $z'$ in $Z^0_\\infty(z_0)$.  We denote by $(x(t),m(t),v(t))$ the blocks of $z(t)$,\n  and we define $(x'(t),m'(t),v'(t))$ similarly.  For all $t>0$, we\n  define $\\hat m(t)\\eqdef m(t)\/(1-e^{-at})$, $\\hat v(t)\\eqdef\n  v(t)\/(1-e^{-bt})$, and we define $\\hat m'(t)$ and $\\hat v'(t)$\n  similarly.  By Prop.~\\ref{prop:adam-bounded}, there exists a compact\n  set $K\\subset \\cZ_+$ s.t.  $ (x(t),\\hat m(t),\\hat v(t))$ and\n  $(x'(t),\\hat m'(t),\\hat v'(t))$ are both in $K$ for all $t> 0$.  We\n  denote by $L_S$ and $L_{\\nabla F}$ the Lipschitz constants of $S$\n  and $\\nabla F$ on the compact set $\\{x:(x,m,v)\\in K\\}$. These\n  constants are finite by Assumptions~\\ref{hyp:F}\n  and \\ref{hyp:S}.\nWe define $M\\eqdef \\sup\\{\\|m\\|_\\infty:(x,m,v)\\in K\\}$.\nDefine $u_x(t) \\eqdef \\|x(t)-x'(t)\\|^2$,\n$u_m(t)\\eqdef \\|\\hat m(t)-\\hat m'(t)\\|^2$ and $u_v(t)\\eqdef \\|\\hat v(t)-\\hat v'(t)\\|^2$.\nLet $\\delta>0$. Define: $u^{(\\delta)}(t) \\eqdef u_x(t)+\\delta u_m(t)+\\delta u_v(t)\\,.$\nBy the chain rule and the Cauchy-Schwarz inequality,\n$\\dot u_x(t)\\leq 2\\|x(t)-x'(t)\\|\\|\\frac{\\hat m(t)}{\\varepsilon +\\sqrt{\\hat v(t)}}-\\frac{\\hat m'(t)}{\\varepsilon +\\sqrt{\\hat v'(t)}}\\|$. Thus,\nusing Lemma~\\ref{lem:v-lowerbound}, there exists $c>0$ s.t.\n\\begin{equation*}\n  \\dot u_x(t)\\leq 2\\|x(t)-x'(t)\\|\\left(\\varepsilon^{-1}\\left\\|\\hat m(t)-\\hat m'(t)\\right\\|\n+\\frac M{2\\varepsilon^2\\sqrt{c\\min(1,t)}}\\left\\|{\\hat v(t)}-{\\hat v'(t)}\\right\\|\\right)\\,.\n\\end{equation*}\nFor any $\\delta>0$,\n$2\\|x(t)-x'(t)\\|\\,\\|\\hat m(t)-\\hat m'(t)\\|\\leq \\delta^{-1\/2}(u_x(t)+\\delta u_m(t))\\leq \\delta^{-1\/2}u^{(\\delta)}(t)$.\nSimilarly, $2\\|x(t)-x'(t)\\|\\,\\|\\hat v(t)-\\hat v'(t)\\|\\leq \\delta^{-1\/2}u^{(\\delta)}(t)$.\nThus, for any $\\delta>0$,\n\\begin{align}\n\\label{eq:ux}\n  \\dot u_x(t)&\\leq \\left(\\frac 1{\\varepsilon\\sqrt \\delta}+\\frac M{2\\varepsilon^2\\sqrt{\\delta c\\min(1,t)}}\\right) u^{(\\delta)}(t)\\,.\n\\end{align}\nWe now study $u_m(t)$. For all $t>0$, we obtain after some algebra:\n$\\frac {d}{dt}\\hat m(t) = a(\\nabla F(x(t)) - \\hat m(t))\/(1-e^{-at})\\,.$\nTherefore,\n$\n\\dot u_m(t) \\leq \\frac{2aL_{\\nabla F}}{1-e^{-at}}\\|\\hat m(t)-\\hat m'(t)\\|\\,\\|x(t) -x'(t)\\|\\,.\n$\nFor any $\\theta>0$, it holds that $2\\|\\hat m(t)-\\hat m'(t)\\|\\,\\|x(t) -x'(t)\\|\\leq \\theta u_x(t)+ \\theta^{-1}u_m(t)$.\nIn particular, letting $\\theta\\eqdef 2L_{\\nabla F}$, we obtain that for all $\\delta>0$,\n\\begin{equation}\n\\resizebox{\\hsize}{!}{$\n\\delta \\dot u_m(t)\\leq  \\frac{a }{2(1-e^{-at})}\\left(4\\delta L_{\\nabla F}^2 u_x(t)+ \\delta u_m(t)\\right)\n                    \\leq  \\left(\\frac a2+\\frac 1{2t}\\right)\\left(4\\delta L_{\\nabla F}^2 u_x(t)+ \\delta u_m(t)\\right)\\,,\n$}\n\\label{eq:um}\n\\end{equation}\nwhere the last inequality is due to the fact that $y\/(1-e^{-y})\\leq 1+y$ for all $y>0$.\nUsing the exact same arguments, we also obtain that\n\\begin{align}\n \\delta \\dot u_v(t)&\\leq  \\left(\\frac b2+\\frac 1{2t}\\right)\\left(4\\delta L_{S}^2 u_x(t)+ \\delta u_m(t)\\right)\\,.\n\\label{eq:uv}\n\\end{align}\nWe now choose any $\\delta$ s.t. $4\\delta \\leq  1\/\\max(L_S^2,L_{\\nabla F}^2)$.\nThen, Eq.~(\\ref{eq:um}) and~(\\ref{eq:uv}) respectively imply that\n$\\delta \\dot u_m(t)\\leq 0.5(a+t^{-1})u^{(\\delta)}(t)$ and\n$\\delta \\dot u_v(t)\\leq 0.5(b+t^{-1})u^{(\\delta)}(t)$.\nSumming these inequalities along with Eq.~(\\ref{eq:ux}), we obtain that for every $t>0$,\n$\\dot u^{(\\delta)}(t) \\leq \\psi(t) u^{(\\delta)}(t)\\,$,\nwhere: $\\psi(t) \\eqdef \\frac{a+b}2+\\frac 1{\\varepsilon\\sqrt \\delta}+\\frac M{2\\varepsilon^2\\sqrt{\\delta c\\min(1,t)}}\n+ \\frac 1t\\,.$\nFrom Gr\\\"onwall's inequality, it holds that for every $t>s>0$,\n$u^{(\\delta)}(t)\\leq u^{(\\delta)}(s)\\exp\\left(\\int_s^t \\psi(s')ds'\\right)\\,$.\nWe first consider the case where $t\\leq 1$. We set $c_1\\eqdef  (a+b)\/2+(\\varepsilon\\sqrt \\delta)^{-1}$\nand $c_2\\eqdef M\/(\\varepsilon^2\\sqrt{\\delta c})$. With these notations,\n$\\int_s^t \\psi(s')ds' \\leq c_1t+c_2\\sqrt t + \\ln \\frac ts\\,.$\nTherefore, $u^{(\\delta)}(t)\\leq \\frac{u^{(\\delta)}(s)}{s}\n\\exp\\left(c_1t+c_2\\sqrt t + \\ln t\\right)\\,$.\nBy Lemma~\\ref{lem:m-v-derivables-en-zero}, recall that $\\dot x(0)$ and\n$\\dot x'(0)$ are both well defined (and coincide). Thus,\n$$\nu_x(s) = \\|x(s)-x'(s)\\|^2\n\\leq  2\\|x(s)-x(0)-\\dot x(0)s\\|^2+2\\|x'(s)-x'(0)-\\dot x'(0)s\\|^2\\,.\n$$\nIt follows that $u_x(s)\/s^2$ converges to zero as $s\\downarrow 0$.\nWe now show the same kind of result for $u_m(s)$ and $u_v(s)$.\nConsider $i\\in \\{1,\\dots,d\\}$. By the mean value theorem, there exists $\\tilde s$ (resp. $\\tilde s'$) in\n$[0,t]$ s.t. $m_i(s)=\\dot m_i(\\tilde s)s$ (resp. $m_i'(s)=\\dot m_i'(\\tilde s')s$).\nThus, $\\hat m_i(s) = \\frac{as}{1-e^{-as}} \\left(\\partial_i F(x(\\tilde s))-m_i(\\tilde s)\\right)$,\nand a similar equality holds for $\\hat m_i'(s)$. Then,\ngiven that $\\|x(\\tilde s)-x'(\\tilde s')\\| \\vee \\|m(\\tilde s)-m'(\\tilde s')\\| \\leq \\|z(\\tilde s)-z'(\\tilde s')\\|$\n, $\\tilde s\\leq s$ and $\\tilde s'\\leq s$,\n$$\n\\frac{|\\hat m_i(s) -\\hat m_i'(s) |}s\n\\leq \\frac{2a(L_{\\nabla F}\\vee 1)s}{1-e^{-as}} \\left(\\frac{\\|z(\\tilde s)-z(0)\\|}{\\tilde s}+\\frac{\\|z'(\\tilde s')-z'(0)\\|}{\\tilde s'}\\right)\\,.\n$$\nBy Lemma~\\ref{lem:m-v-derivables-en-zero}, $z$ and $z'$ are differentiable at point zero.\nThen, the above inequality gives $\\limsup_{s\\downarrow 0}\\frac{|\\hat m_i(s) -\\hat m_i'(s) |}s \\leq 4(L_{\\nabla F}\\vee 1)\\|\\dot z(0)\\|$\nand\n\n\\limsup_{s\\downarrow 0}\\frac{u_m(s)}{s^2} \\leq 16d(L_{\\nabla F}^2\\vee 1)\\|\\dot z(0)\\|^2\\,.\n\nTherefore, $u_m(s)\/s$ converges to zero as $s\\downarrow 0$.\nBy similar arguments, it can be shown that\n$\\limsup_{s\\downarrow 0}{u_v(s)}\/{s^2} \\leq 16d(L_{S}^2\\vee 1)\\|\\dot z(0)\\|^2$,\nthus $\\lim u_v(s)\/s=0$.\nFinally, we obtain that\n${u^{(\\delta)}(s)}\/{s}$ converges to zero as $s\\downarrow 0$.\nLetting $s$ tend to zero, we obtain that for every\n$t\\leq 1$, $u^{(\\delta)}(t)=0$. Setting $s=1$ and $t>1$,\nand noting that $\\psi$ is integrable on $[1,t]$, it follows that $u^{(\\delta)}(t)=0$ for all $t>1$.\nThis proves that $z=z'$.\n\\end{proof}\n\nWe recall that a semiflow $\\Phi$ on the space $(E,\\sd)$ is a continuous map\n$\\Phi$ from $[0,+\\infty)\\times E$ to $E$ defined by $(t,x) \\mapsto \\Phi(t,x) = \\Phi_t(x)$\nsuch that $\\Phi_0$ is the identity and $\\Phi_{t+s} = \\Phi_t\\circ\\Phi_s$ for all $(t,s)\\in [0,+\\infty)^2$.\n\n\\begin{proposition}\n  \\label{prop:semiflow}\nLet Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\nAssume that $0< b\\leq 4a$.\nThe map $Z_{\\infty}^{\\infty}$ is single-valued on $\\cZ_+\\to C([0,+\\infty),\\cZ_+)$\n\\emph{i.e.}, there exists a unique global solution to~(\\ref{eq:ode-a})\nstarting from any given point in $\\cZ_+$.\nMoreover, the following map is a semiflow:\n\\begin{equation}\n  \\label{eq:flot}\n  \\begin{array}[h]{rcl}\n    \\Phi:[0,+\\infty)\\times \\cZ_+&\\to& \\cZ_+ \\\\\n(t,z) &\\mapsto& Z_{\\infty}^{\\infty}(z)(t)\n  \\end{array}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe result is a direct consequence of Lemma~\\ref{prop:unique-adam}.\n\\end{proof}\n\n\\subsection{Proof of Th.~\\ref{th:cv-adam}}\n\\label{sec:convergence}\n\\subsubsection{Convergence of the semiflow}\n\nWe first recall some useful definitions and results.\nLet $\\Psi$ represent any semiflow on an arbitrary metric space $(E,\\sd)$.\nA point $z\\in E$ is called an \\emph{equilibrium point} of the semiflow $\\Psi$ if $\\Psi_t(z)=z$ for all $t\\geq 0$.\nWe denote by $\\Lambda_\\Psi$ the set of equilibrium points of~$\\Psi$.\nA continuous function $\\sV:E\\to\\bR$ is called a \\emph{Lyapunov function} for the semiflow $\\Psi$\nif $\\sV(\\Psi_t(z))\\leq \\sV(z)$ for all $z\\in E$ and all $t\\geq 0$.\nIt is called a \\emph{strict Lyapunov function} if, moreover,\n$\n\\{ z\\in E\\,:\\, \\forall t\\geq 0,\\,\\sV(\\Psi_t(z))=\\sV(z) \\}= \\Lambda_\\Psi\n$.\nIf $\\sV$ is a strict Lyapunov function for $\\Psi$ and if $z\\in E$ is a point s.t. $\\{\\Psi_t(z):t\\geq 0\\}$ is relatively compact,\nthen it holds that $\\Lambda_\\Psi\\neq \\emptyset$ and $\\sd(\\Psi_t(z),\\Lambda_\\Psi)\\to 0$, see \\cite[Th.~2.1.7]{haraux1991systemes}.\nA continuous function $z:[0,+\\infty)\\to E$ is said to be an asymptotic pseudotrajectory (APT)\nfor the semiflow $\\Psi$ if for every $T\\in (0,+\\infty)$,\n$\n\\lim_{t\\to+\\infty} \\sup_{s\\in [0,T]} \\sd(z(t+s),\\Psi_s(z(t))) = 0\\,.\n$\nThe following result follows from \\cite[Th.~5.7]{ben-(cours)99} and \\cite[Prop.~6.4]{ben-(cours)99}.\n\\begin{proposition}[\\cite{ben-(cours)99}]\\hfill\\\\\n\\label{prop:benaim}\n  Consider a semiflow $\\Psi$ on $(E,d)$ and a map $z:[0,+\\infty)\\to E$. Assume the following:\n  \\begin{enumerate}[{\\it i)}]\n  \\item $\\Psi$ admits a strict Lyapunov function $\\sV$.\n  \\item The set $\\Lambda_\\Psi$ of equilibrium points of $\\Psi$ is compact.\n  \\item  $\\sV(\\Lambda_\\Psi)$ has an empty interior.\n  \\item $z$ is an APT of $\\Psi$.\n  \\item $z([0,\\infty))$ is relatively compact.\n  \\end{enumerate}\nThen, $\n\\bigcap_{t\\geq 0}\\overline{z([t,\\infty))}$ is a compact connected subset of $\\Lambda_\\Psi$\\,.\n\\end{proposition}\n\nFor every $\\delta>0$ and every $z = (x,m,v)\\in \\cZ_+$, define:\n\\begin{equation}\nW_\\delta(x,m,v) \\eqdef V_{\\infty}(x,m,v) - \\delta \\ps{\\nabla F(x),m} + \\delta \\|S(x)-v\\|^2\\,,\\label{eq:Wdelta}\n\\end{equation}\nwhere we recall that $V_\\infty(z)\\eqdef \\lim_{t\\to \\infty} V(t,z)$ for every $z\\in \\cZ_+$ and $V$ is defined by Eq.(\\ref{eq:V}).\nConsider the set $\\cE\\eqdef h_\\infty^{-1}(\\{0\\})$ of all equilibrium points of (\\ref{eq:ode-a}), namely:\n$\\cE = \\{(x,m,v)\\in \\cZ_+:\\nabla F(x)=0,m=0,v=S(x)\\}\\,$.\nThe set $\\cE$ is non-empty by Assumption~\\ref{hyp:coercive}.\n\n\\begin{proposition}\n  \\label{prop:Wstrict}\nLet Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\nAssume that $0< b\\leq 4a$.\nLet $K\\subset \\cZ_+$ be a compact set. Define $K'\\eqdef \\overline{\\{\\flot(t,z):t\\geq 0, z\\in K\\}}$.\nLet $\\bflot:[0,+\\infty)\\times K'\\to K'$ be the restriction of the semiflow $\\flot$ to $K'$ \\emph{i.e.},\n$\\bflot(t,z) = \\flot(t,z)$ for all  $t\\geq 0, z\\in K'$. Then,\n\\begin{enumerate}[{\\it i)}]\n\\item $K'$ is compact.\n\\item $\\bflot$ is well defined and is a semiflow on $K'$.\n\\item The set of equilibrium points of $\\bflot$ is equal to $\\cE\\cap K'$.\n\\item There exists $\\delta>0$ s.t. $W_\\delta$ is a strict Lyapunov function for the semiflow $\\bflot$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nThe first point is a consequence of Prop.~\\ref{prop:bounded}.\nThe second point stems from Prop.~\\ref{prop:semiflow}.\nThe third point is immediate from the definition of $\\cE$ and the fact that $\\bflot$ is valued in $K'$.\nWe now prove the last point.\nConsider $z\\in K'$ and write $\\bflot_t(z)$ under the form $\\bflot_t(z) = (x(t),m(t),v(t))$. For \\emph{any} map ${\\mathsf W}:\\cZ_+\\to\\bR$, define\nfor all $t>0$, $ \\cL_{\\mathsf W}(t) \\eqdef \\limsup_{s\\to 0} s^{-1}({\\mathsf W}(\\bflot_{t+s}(z)) -   {\\mathsf W}(\\bflot_{t}(z)))\\,.$\nIntroduce $G(z)\\eqdef -\\ps{\\nabla F(x),m}$ and $H(z)\\eqdef \\|S(x)-v\\|^2$ for every $z=(x,m,v)$.\nConsider $\\delta>0$ (to be specified later on).\nWe study $\\cL_{W_\\delta} = \\cL_V + \\delta \\cL_{G} + \\delta \\cL_{H}$.\nNote that $\\bflot_t(z)\\in K'\\cap \\cZ_+^*$ for all $t>0$ by Lemma~\\ref{lem:v-positif}. Thus, $t\\mapsto V_\\infty(\\bflot_t(z))$\nis differentiable at any point $t>0$ and the derivative coincides with $\\cL_V(t) = \\dot V_\\infty(\\bflot_t(z))$.\nDefine $C_1\\eqdef \\sup\\{\\|v\\|_\\infty:(x,m,v)\\in K'\\}$.\nThen, by Lemma~\\ref{lem:V}, $\\cL_V(t)\\leq -\\varepsilon(\\varepsilon+\\sqrt{C_1})^{-2} \\left\\|m(t)\\right\\|^2$.\nLet $L_{\\nabla F}$ be the Lipschitz constant of $\\nabla F$\non $\\{x:(x,m,v)\\in K'\\}$.\nFor every $t>0$,\n\\begin{align*}\n  \\cL_G(t\n&\\leq  \\limsup_{s\\to 0} s^{-1}\\|\\nabla F(x(t))- \\nabla F(x(t+s))\\| \\|m(t+s)\\| - \\ps{\\nabla F(x(t)),\\dot m(t)}\\\\\n&\\leq  L_{\\nabla F}\\varepsilon^{-1}\\|m(t)\\|^2 - a\\|\\nabla F(x(t))\\|^2  + a\\ps{\\nabla F(x(t)),m(t)}\\\\\n&\\leq - \\frac a2\\|\\nabla F(x(t))\\|^2  +  \\left(\\frac a2+\\frac{L_{\\nabla F}}{\\varepsilon}\\right)\\|m(t)\\|^2\\,.\n\\end{align*}\nDenote by $L_S$ the Lipschitz constant of $S$ on $\\{x:(x,m,v)\\in K'\\}$.\nFor every $t>0$,\n\\begin{eqnarray*}\n  \\cL_H(t\n&=&  \\limsup_{s\\to 0} s^{-1}(\\|S(x(t+s))-S(x(t)) + S(x(t))-v(t+s)\\|^2-\\|S(x(t))-v(t)\\|^2)\\\\\n&=& - 2\\ps{S(x(t))-v(t),\\dot v(t)\n+\\limsup_{s\\to 0} 2s^{-1}\\ps{S(x(t+s))-S(x(t)),S(x(t))-v(t+s)} \\\\\n&\\leq&   - 2b\\|S(x(t))-v(t)\\|^2+2L_S\\varepsilon^{-1} \\|m(t)\\| \\|S(x(t))-v(t)\\|\\,.\n\\end{eqnarray*}\nUsing that $2 \\|m(t)\\| \\|S(x(t))-v(t)\\|\\leq \\frac {L_S}{b\\varepsilon}\\|m(t)\\|^2 + \\frac {b\\varepsilon}{L_S}\\|S(x(t))-v(t)\\|^2$, we obtain\n$\n\\cL_H(t) \\leq - b\\|S(x(t))-v(t)\\|^2+\\frac {L_S^2}{b\\varepsilon^2}\\|m(t)\\|^2\\,.\n$\nHence,\nfor every $t>0$,\n$$\n  \\cL_{W_\\delta}(t) \\leq -M(\\delta)\n  \\|m(t)\\|^2  - \\frac {a\\delta}2\\|\\nabla\n  F(x(t))\\|^2 - \\delta b \\|S(x(t))-v(t)\\|^2\\,.\n$$\nwhere $M(\\delta)\\eqdef \\varepsilon(\\varepsilon+\\sqrt{C_1})^{-2} -\\frac {\\delta\n      L_S^2}{b\\varepsilon^2} - \\delta \\left(\\frac a2+\\frac{L_{\\nabla\n        F}}{\\varepsilon}\\right)\\,.$\nChoosing $\\delta$ s.t. $M(\\delta)>0$,\n\\begin{equation}\n\\forall t>0,\\ \\  \\cL_{W_\\delta}(t) \\leq -c\\left( \\|m(t)\\|^2 + \\|\\nabla  F(x(t))\\|^2 + \\|S(x(t))-v(t)\\|^2\\right)\\,,\\label{eq:ae-derivee}\n\\end{equation}\nwhere $c \\eqdef \\min\\{ M(\\delta), \\frac {a\\delta}2, \\delta b\\}$.\nIt can easily be seen that for every $z\\in K'$,  $t\\mapsto W_\\delta(\\bflot_t(z))$ is Lipschitz continuous, hence absolutely continuous.\nIts derivative almost everywhere coincides with $\\cL_{W_\\delta}$, which is non-positive.\nThus, $W_\\delta$ is a Lyapunov function for $\\bflot$.\nWe prove that the Lyapunov function is strict.\nConsider $z\\in K'$ s.t. $W_\\delta(\\bflot_t(z))=W_\\delta(z)$ for all $t>0$.\nThe derivative almost everywhere of  $t\\mapsto W_\\delta(\\bflot_t(z))$ is identically zero,\nand by Eq. (\\ref{eq:ae-derivee}), this implies that\n $-c\\left(\\|m_t\\|^2 + \\|\\nabla F(x_t)\\|^2 +\\|S(x_t)-v_t\\|^2\\right)$ is equal to zero\nfor every $t$ a.e. (hence, for every~$t$, by continuity of $\\bflot$).\nIn particular for $t=0$, $m=\\nabla F(x)=0$ and $S(x)-v=0$.\nHence, $z\\in h_\\infty^{-1}(\\{0\\})$.\n\\end{proof}\n\n\\begin{corollary}\n  \\label{coro:cv}\nLet Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold.\nAssume that $0< b\\leq 4a$.\nFor every $z\\in \\cZ_+$, $\\lim_{t\\to \\infty} \\sd(\\Phi(z,t),\\cE)=0\\,.$\n\\end{corollary}\n\\begin{proof}\nUse Prop.~\\ref{prop:Wstrict} with $K\\eqdef\\{z\\}$.\nand \\cite[Th.~2.1.7]{haraux1991systemes}.\n\\end{proof}\n\n\n\\subsubsection{Asymptotic Behavior of the Solution to (\\ref{eq:ode})}\n\\label{sec:cv-non-autonomous}\n\n\n\\begin{proposition}[APT]\n \n  Let Assumptions~\\ref{hyp:coercive}, \\ref{hyp:S>0}, \\ref{hyp:F} and \\ref{hyp:S} hold true.\n  Assume that $0< b\\leq 4a$.\nThen, for every $z_0\\in \\cZ_0$, $Z^0_\\infty(z_0)$ is an asymptotic pseudotrajectory\nof the semiflow  $\\Phi$ given by~(\\ref{eq:flot}).\n\\label{prop:apt}\n\\end{proposition}\n\\begin{proof}\n  Consider  $z_0\\in \\cZ_0$, $T\\in (0,+\\infty)$ and define $z\\eqdef Z_\\infty^0(z_0)$.\nConsider $t\\geq 1$. For every $s\\geq 0$, define $\\Delta_t(s)\\eqdef  \\|z(t+s)-\\flot(z(t))(s)\\|$.\nThe aim is to prove that $\\sup_{s\\in [0,T]}\\Delta_t(s)$ tends to zero as~$t\\to\\infty$.\nPutting together Prop.~\\ref{prop:adam-bounded} and Lemma~\\ref{lem:v-lowerbound},\nthe set $K\\eqdef \\overline{\\{z(t):t\\geq 1\\}}$\nis a compact subset of $\\cZ_+^*$.\nDefine $C(t)\\eqdef \\sup_{s\\geq 0}\\sup_{z'\\in K}\\|h(t+s,z')-h_\\infty(z')\\|$.\nIt can be shown that $\\lim_{t\\to\\infty} C(t)=0$.\nWe obtain that for every $s\\in [0,T]$, $\n\\Delta_t(s)\\leq T C(t)  + \\int_0^s\n  \\|h_\\infty(z(t+s'))-h_{\\infty}(\\flot(z(t))(s'))\\|ds'\\,.$\nBy Lemma~\\ref{lem:v-lowerbound}, $K'\\eqdef \\overline{\\bigcup_{z'\\in\\flot(K)}z'([0,\\infty))}$ is a compact subset of $\\cZ_+^*$.\nIt is immediately seen from the definition that $h_{\\infty}$ is Lipschitz continuous on every compact subset of $\\cZ_+^*$, hence on $K\\cup K'$. Therefore, there exists a constant $L$, independent from $t,s$, s.t.\n$\n\\Delta_t(s)\\leq T C(t)  + \\int_0^s L \\Delta_t(s')ds'\\qquad (\\forall t\\geq 1, \\forall s\\in[0,T])\\,.\n$\nUsing Gr\\\"onwall's lemma, it holds that for all $s\\in [0,T]$,\n$\n\\Delta_t(s)\\leq TC(t) e^{Ls}\\,.\n$\nAs a consequence,\n$\\sup_{s\\in [0,T]}\\Delta_t(s)\\leq TC(t) e^{LT}$ and the righthand side converges to zero\nas $t\\to\\infty$.\n\\end{proof}\n\n\\subsubsection*{End of the Proof of Th.~\\ref{th:cv-adam}}\n\nBy Prop.~\\ref{prop:adam-bounded}, the set $K\\eqdef \\overline{Z^0_\\infty(z_0)([0,\\infty))}$ is a compact subset of $\\cZ_+$.\nDefine $K'\\eqdef \\overline{\\{\\flot(t,z):t\\geq 0, z\\in K\\}}\\,,$ and let\n $\\bflot:[0,+\\infty)\\times K'\\to K'$ be the restriction $\\flot$ to $K'$.\nBy Prop.~\\ref{prop:Wstrict}, there exists $\\delta>0$ s.t.\n$W_\\delta$ is  a strict Lyapunov function for the semiflow $\\bflot$.\nMoreover, the set of equilibrium points coincides with $\\cE\\cap K'$.\nIn particular, the equilibrium points of $\\bflot$ form a compact set.\nBy Prop.~\\ref{prop:apt}, $Z^0_\\infty(z_0)$ is an APT of $\\bflot$.\nNote that every $z\\in\\cE$ can be written under the form\n$z=(x,0,S(x))$ for some $x\\in \\cS$.\nFrom the definition of $W_\\delta$ in (\\ref{eq:Wdelta}),\n$W_\\delta(z)=W_\\delta(x,0,S(x)) =V_{\\infty}(x,0,S(x)) = F(x)$.\nSince $F(\\cS)$ is assumed to have an empty interior, the same holds\nfor $W_\\delta(\\cE\\cap K')$. By Prop.~\\ref{prop:benaim},\n$\n\\bigcap_{t\\geq 0}\\overline{Z^0_\\infty(z_0)([t,\\infty))} \\subset \\cE\\cap K'\\,.\n$\nThe set in the righthand side coincides with the set of limits of convergent\nsequences of the form $Z^0_\\infty(z_0)(t_n)$ for $t_n\\to\\infty$.\nAs $Z^0_\\infty(z_0)([0,\\infty))$ is a bounded set, $\\sd(Z^0_\\infty(z_0)(t),\\cE)$ tends to zero.\n\n\n\\subsection{Proof of Th.~\\ref{thm:asymptotic_rates}}\n\\label{sec:cont_asymptotic_rates}\n\nThe proof follows the path of \\cite[Th.~10.1.6, Th.~10.2.3]{harauxjendoubi2015},\nbut requires specific adaptations to deal with the dynamical system at hand.\nDefine for all $\\delta>0$, $t>0$, and $z=(x,m,v)$,\n\\begin{equation}\n\\tilde W_\\delta(t,(x,m,v)) \\eqdef V(t,(x,m,v)) - \\delta \\ps{\\nabla F(x),m} + \\delta \\|S(x)-v\\|^2\\,.\\label{eq:Wdelta-t}\n\\end{equation}\nThe function $\\tilde W_\\delta$ is the non-autonomous version of the function (\\ref{eq:Wdelta}).\nConsider a fixed $x_0\\in \\bR^d$, and define $w_{\\delta}(t)\\eqdef \\tilde W_\\delta(t,z(t))$\nwhere $z(t)=(x(t),m(t),v(t))$ is the solution to~(\\ref{eq:ode}) with initial condition $(x_0,0,0)$.\nThe proof uses the following steps.\n\n\\begin{enumerate}[{\\sl i)},leftmargin=11pt]\n\\item \\textit{Upper-bound on  $w_\\delta(t)$.}\nFrom Eq.~(\\ref{eq:V}),\nwe obtain that for every $t\\geq 1$,\n\nV(t,z(t))\\leq |F(x(t))|+\\frac {\\|m(t)\\|^2}{2a\\varepsilon (1-e^{-a})}\\,.\n\nUsing $\\ps{\\nabla F(x),m}\\leq (\\|\\nabla F(x)\\|^2+\\|m\\|^2)\/2$, we obtain that there exists a constant\n$c_1$ (depending on $\\delta$) s.t. for every $t\\geq 1$,\n\\begin{equation}\n  \\label{eq:w-up}\n  w_\\delta(t) \\leq c_1\\left(|F(x(t))|+\\|m(t)\\|^2+\\|\\nabla F(x(t))\\|^2+ \\|S(x(t))-v(t)\\|^2\\right)\\,.\n\\end{equation}\n\n\\item \\textit{Upper-bound on  $\\frac d{dt} w_\\delta(t)$.}\nThe function $w_{\\delta}$ is absolutely continuous on $[1,+\\infty)$.\n   \n   \n   \n   \n   \n   \nMoreover, there exist $\\delta>0$, $c_2>0$ (both depending on\n$x_0$) s.t. for every $t\\geq 1$ a.e.,\n\\begin{equation}\n  \\label{eq:dw-up}\n  \\frac d{dt} w_\\delta (t) \\leq -c_2\\left(\\|m(t)\\|^2+\\|\\nabla F(x(t))\\|^2+\\|S(x(t))-v(t)\\|^2\\right)\\,.\n\\end{equation}\nThe proof of Eq.~(\\ref{eq:dw-up}) uses arguments that are similar to\nthe ones used in the proof of Prop.~\\ref{prop:Wstrict} (just\nuse Lemma~\\ref{lem:V} to bound the derivative of the first term in\nEq.~(\\ref{eq:Wdelta-t})).  For this reason, it is omitted.\n\\item \\textit{Positivity of $w_\\delta(t)$.} By Lemma~\\ref{lem:V},\nthe function $t\\mapsto V(t,z(t))$ is decreasing. As it is lower bounded,\n$\\ell\\eqdef \\lim_{t\\to\\infty}V(t,z(t))$ exists. By Th.~\\ref{th:cv-adam},\n$m(t)$ tends to zero, hence this limit coincides with $\\lim_{t\\to\\infty} F(x(t))$.\nReplacing $F$ with $F-\\ell$, one can assume without loss of generality that $\\ell=0$.\nBy Eq.~(\\ref{eq:dw-up}), $w_\\delta$ is non-increasing on $[1,+\\infty)$, hence converging\nto some limit. Using again Th.~\\ref{th:cv-adam}, $\\ps{\\nabla F(x(t)),m(t)}\\to 0$\nand $S(x(t))-v(t)\\to 0$. Thus, $\\lim_{t\\to\\infty} w_\\delta(t) = \\ell = 0$.\nAssume that there exists $t_0\\geq 1$ s.t. $w_\\delta(t_0)=0$. Then, $w_\\delta$\nis constant on $[t_0,+\\infty)$. By Eq.~(\\ref{eq:dw-up}), this implies that $m(t)=0$ on this interval.\nHence, $d x(t)\/dt = 0$. This means that $x(t) = x(t_0)$ for all $t\\geq t_0$. By Th.~\\ref{th:cv-adam},\nit follows that $x(t_0)\\in \\cS$. In that case, the final result is shown. Therefore, one\ncan assume that $w_\\delta(t)>0$ for all $t\\geq 1$.\n\n\\item \\textit{Putting together (\\ref{eq:w-up}) and (\\ref{eq:dw-up}) using the \\L{}ojasiewicz condition.}\nBy Prop. \\ref{prop:benaim} and \\ref{prop:apt}, the set\n$\nL\\eqdef \\overline{\\bigcup_{s\\geq 0}\\{z(t):t\\geq s\\}}\\,\n$\nis a compact connected subset of $\\cE = \\{(x,0,S(x)):\\nabla F(x)=0\\}$.\nThe set $\\mathcal{U}\\eqdef \\{x:(x,0,S(x))\\in L\\}$ is a compact and connected subset of $\\cS$.\nUsing Assumption~\\ref{hyp:lojasiewicz_prop} and \\cite[Lemma 2.1.6]{harauxjendoubi2015},\nthere exist constants $\\sigma,c>0$ and $\\theta\\in (0,\\frac 12]$, s.t.\n$\\|\\nabla F(x)\\| \\geq c|F(x)|^{1-\\theta}\\,$ for all $x$ s.t.~$\\sd(x,\\mathcal{U})<\\sigma\\,$.\nAs $\\sd(x(t), \\mathcal{U})\\to 0$, there exists $T\\geq 1$ s.t. for all $t\\geq T$,\n$\\|\\nabla F(x(t))\\|\\geq c |F(x(t))|^{1-\\theta}$. Thus, we may replace the term $\\|\\nabla F(x(t))\\|^2$\nin the righthand side of Eq.~(\\ref{eq:dw-up}) using\n$\\|\\nabla F(x(t))\\|^2\\geq \\frac 12\\|\\nabla F(x(t))\\|^2+ \\frac 12|F(x(t))|^{2(1-\\theta)}$.\nUpon noting that $2(1-\\theta)\\geq 1$,\nwe thus obtain that there exists a constant $c_3$ and some $T'\\geq 1$ s.t. for $t\\geq T'$ a.e.,\n$$\n\\frac d{dt} w_\\delta (t) \\leq -c_3\\left(\\|m(t)\\|^2+\\|\\nabla F(x(t))\\|^2+|F(x(t))|+\\|S(x(t))-v(t)\\|^2\\right)^{2(1-\\theta)}\\,.\n$$\nPutting this inequality together with Eq.~(\\ref{eq:w-up}), we obtain that for some constant $c_4>0$ and for all\n$t\\geq T'$ a.e.,\n$\n\\frac d{dt} w_\\delta (t) \\leq -c_4 w_\\delta(t)^{2(1-\\theta)}\\,.\n$\n\\item \\textit{End of the proof.} Following the arguments of \\cite[Th.~10.1.6]{harauxjendoubi2015},\nby integrating the preceding inequality,\nover $[T',t]$, we obtain\n$\nw_\\delta(t) \\leq c_5 t^{-\\frac 1{1-2\\theta}\n$\nfor $t\\geq T'$\nin the case where $\\theta<\\frac 12$, whereas $w_\\delta(t)$ decays exponentially if $\\theta=\\frac 12$.\nFrom now on, we focus on the case $\\theta<\\frac 12$.\nBy definition of~(\\ref{eq:ode}), $\\|\\dot{x}(t)\\|^2\\leq \\|m(t)\\|^2\/((1-e^{-a T'})^2\\varepsilon^2)$\nfor all $t\\geq T'$. Since Eq.~(\\ref{eq:dw-up}) implies $\\|m(t)\\|^2\\leq -\\dot w_\\delta(t)\/c_2$,\nwe deduce that there exists $c,c'>0$ s.t. for all $t\\geq T'$,\n\n \n  \\int_t^{2t} \\|\\dot{x}(s)\\|^2 ds \\leq c w_\\delta(t) \\leq c' t^{-\\frac 1{1-2\\theta}}\\,.\n\nApplying \\cite[Lemma 2.1.5]{harauxjendoubi2015}, it follows that\n$\\int_t^{\\infty} \\|\\dot{x}(s)\\|^2 ds  \\leq c  t^{-\\frac{\\theta}{1-2\\theta}}$\nfor some other constant $c$.\nTherefore $x^* \\eqdef \\lim_{t \\to +\\infty} x(t)$ exists by Cauchy's criterion and\nfor all $t\\geq T'$, $\\|x(t)-x^*\\| \\leq c  t^{-\\frac{\\theta}{1-2\\theta}}$\\,.\nFinally, since $x(t)\\to a$, we remark that, using the same arguments,\nthe global \\L{}ojasiewicz exponent $\\theta$ can be replaced by any \\L{}ojasiewicz exponent\nof $f$ at $x^*$.\nWhen $\\theta=\\frac 12$, the proof follows the same line.\n\\end{enumerate}\n\n\n\\section{Proofs of Section~\\ref{sec:discrete}}\n\\label{sec:proofs_sec_discrete}\n\\subsection{Proof of Th.~\\ref{th:weak-cv}}\n\nGiven an initial point $x_0\\in \\bR^d$ and a stepsize $\\gamma>0$,\nwe consider the iterates $z_n^{\\gamma}$ given by~(\\ref{eq:znT})\nand $z_0^\\gamma\\eqdef (x_0,0,0)$.\nFor every $n\\in \\bN^*$ and every $z\\in \\cZ_+$, we define\n\\begin{equation*}\nH_\\gamma(n,z,\\xi)\\eqdef \\gamma^{-1} (T_{\\gamma,\\bar\\alpha(\\gamma),\\bar\\beta(\\gamma)}(n,z,\\xi)-z)\\,.\n\\end{equation*}\nThus, $z_n^\\gamma = z_{n-1}^\\gamma+\\gamma H_\\gamma(n,z_{n-1}^\\gamma,\\xi_n)$ for every $n\\in\\bN^*$.\nFor every $n\\in\\bN^*$ and every $z\\in \\cZ$ of the form\n$z=(x,m,v)$, we define $e_\\gamma(n,z)\\eqdef (x,(1-\\bar\\alpha(\\gamma)^n)^{-1}m,(1-\\bar\\beta(\\gamma)^n)^{-1}v)$,\nand set $e_\\gamma(0,z)\\eqdef z$.\n\\begin{lemma}\n  \\label{lem:moment-H}\nLet Assumptions~\\ref{hyp:model}, \\ref{hyp:alpha-beta} and \\ref{hyp:moment-f} hold true.\nThere exists $\\bar \\gamma_0>0$ s.t. for every $R>0$, there exists $s>0$,\n\\begin{equation}\n\\sup\\left\\{\\bE\\left(\\left\\|H_\\gamma(n+1,z,\\xi)\\right\\|^{1+s}\\right):\\gamma\\in (0,\\bar \\gamma_0], n\\in \\bN, z\\in \\cZ_+\\,\\text{s.t.}\\,\\|e_\\gamma(n,z)\\|\\leq R\\right\\}<\\infty\\,.\n\\label{eq:UI}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\n\n\nLet $R>0$.\nWe denote by $(H_{\\gamma,\\sx},H_{\\gamma,\\sm},H_{\\gamma,\\sv})$ the block components of $H_\\gamma$.\nThere exists a constant $C_s$ depending only on $s$ s.t.\n$\\|H_\\gamma\\|^{1+s}\\leq C_s(\\|H_{\\gamma,\\sx}\\|^{1+s}+\\|H_{\\gamma,\\sm}\\|^{1+s}+\\|H_{\\gamma,\\sv}\\|^{1+s})$.\nHence, it is sufficient to prove that Eq.~(\\ref{eq:UI}) holds\nrespectively when replacing $H_\\gamma$ with each of $H_{\\gamma,\\sx},H_{\\gamma,\\sm},H_{\\gamma,\\sv}$.\nConsider $z=(x,m,v)$ in $\\cZ_+$. We write:\n$\\|H_{\\gamma,\\sx}(n+1,z,\\xi)\\| \\leq \\varepsilon^{-1}(\\|\\frac{m}{1-\\bar \\alpha(\\gamma)^n}\\|+\\|\\nabla f(x,\\xi)\\|)\\,.$\nThus, for every $z$ s.t. $\\|e_\\gamma(n,z)\\|\\leq R$, there exists a constant $C$ depending only on $\\varepsilon$, $R$ and $s$ s.t.\n$\\|H_{\\gamma,\\sx}(n+1,z,\\xi)\\|^{1+s} \\leq C(1+ \\|\\nabla f(x,\\xi)\\|^{1+s})$. By Assumption~\\ref{hyp:moment-f}, (\\ref{eq:UI}) holds for\n$H_{\\gamma,\\sx}$ instead of $H_\\gamma$. Similar arguments hold for $H_{\\gamma,\\sm}$ and $H_{\\gamma,\\sv}$\nupon noting that the functions $\\gamma\\mapsto(1-\\bar \\alpha(\\gamma))\/\\gamma$ and $\\gamma\\mapsto (1-\\bar \\beta(\\gamma))\/\\gamma$\nare bounded under Assumption~\\ref{hyp:alpha-beta}.\n\\end{proof}\n\nFor every $R>0$, and every arbitrary sequence $z=(z_n:n\\in\\bN)$ on $\\cZ_+$, we define\n$\\tau_R(z) \\eqdef \\inf\\{n\\in \\bN:\\|e_\\gamma(n,z_n)\\|>R\\}$ with the convention that $\\tau_R(z)=+\\infty$ when the set is empty.\nWe define the map $B_R:\\cZ_+^\\bN\\to\\cZ_+^\\bN$ given for any arbitrary sequence  $z=(z_n:n\\in\\bN)$ on $\\cZ_+$ by\n$B_R(z)(n) = z_n\\1_{n<\\tau_R(z)}+z_{\\tau_R(z)}\\1_{n\\geq\\tau_R(z)}$.\nWe define the random sequence $z^{\\gamma,R}\\eqdef B_R(z^\\gamma)$. Recall that a family $(X_i:i\\in I)$ of random variables on some Euclidean space\nis called \\emph{uniformly integrable} if $\\lim_{A\\to+\\infty} \\sup_{i\\in I}\\bE(\\|X_i\\|\\1_{\\|X_i\\|>A})=0$.\n\\begin{lemma}\n  \\label{lem:UI}\nLet Assumptions~\\ref{hyp:model}, \\ref{hyp:alpha-beta}, \\ref{hyp:moment-f} and \\ref{hyp:iid} hold true.\nThere exists $\\bar \\gamma_0>0$ s.t. for every $R>0$, the family of r.v.\n$(\\gamma^{-1}(z_{n+1}^{\\gamma,R} - z_n^{\\gamma,R}):n\\in \\bN,\\gamma\\in (0,\\bar \\gamma_0])$ is uniformly integrable.\n\\end{lemma}\n\\begin{proof}\nLet $R>0$.\nAs the event $\\{n<\\tau_R(z^\\gamma)\\}$ coincides with\n$\\bigcap_{k=0}^n\\{\\|e_\\gamma(k,z_k^\\gamma)\\|\\leq R\\}$, it holds that\nfor every $n\\in \\bN$,\n\\begin{equation*}\n  \\frac{z_{n+1}^{\\gamma,R} - z_n^{\\gamma,R}}\\gamma =\\frac{z_{n+1}^{\\gamma} - z_n^{\\gamma}}\\gamma \\1_{n<\\tau_R(z^\\gamma)}\n = H_\\gamma(n+1,z_n^\\gamma,\\xi_{n+1}) \\prod_{k=0}^n\\1_{\\|e_\\gamma(k, z_k^\\gamma)\\|\\leq R} \\,.\n\\end{equation*}\nChoose $\\bar \\gamma_0>0$ and $s>0$ as in Lemma~\\ref{lem:moment-H}.\nFor every $\\gamma\\leq \\bar \\gamma_0$,\n\\begin{equation*}\n  \\resizebox{\\hsize}{!}{$\n  \\bE\\left(\\left\\|\\gamma^{-1}(z_{n+1}^{\\gamma,R} - z_n^{\\gamma,R})\\right\\|^{1+s}\\right)\n \\leq \\sup\\left\\{ \\bE\\left(\\left\\|H_{\\gamma'}(\\ell+1,z,\\xi)\\right\\|^{1+s} \\right):\\gamma'\\in (0,\\bar \\gamma_0],\\ell\\in \\bN,z\\in \\cZ_+, \\|e_\\gamma(\\ell,z)\\|\\leq R\\right\\}\\,.\n $}\n\\end{equation*}\nBy Lemma~\\ref{lem:moment-H}, the righthand side is finite and does not depend on $(n,\\gamma)$.\n\\end{proof}\nFor a fixed $\\gamma>0$, we define the interpolation map $\\sX_\\gamma:\\cZ^\\bN\\to C([0,+\\infty),\\cZ)$ as follows for every\nsequence $z=(z_n:n\\in \\bN)$ on $\\cZ$:\n$$\n  \\sX_\\gamma(z)\\,:t \\mapsto z_{\\lfloor \\frac t\\gamma\\rfloor} + (t\/\\gamma-\\lfloor t\/\\gamma\\rfloor)(z_{\\lfloor \\frac t\\gamma\\rfloor+1}-z_{\\lfloor \\frac t\\gamma\\rfloor})\\,.\n$$\nFor every $\\gamma,R>0$, we define $\\sz^{\\gamma,R}\\eqdef \\sX_\\gamma(z^{\\gamma,R}) = \\sX_\\gamma\\circ B_R (z^\\gamma)$.\nNamely, $\\sz^{\\gamma,R}$ is the interpolated process associated with the sequence $(z^{\\gamma,R}_n)$.\nIt is a random variable on $C([0,+\\infty),\\cZ)$.\nWe recall that $\\cF_n$ is the $\\sigma$-algebra generated by the r.v. $(\\xi_k:1\\leq k\\leq n)$.\nFor every $\\gamma,n,R$, we use the notation: $\\Delta_0^{\\gamma,R}\\eqdef 0$ and\n$$\n\\Delta_{n+1}^{\\gamma,R} \\eqdef \\gamma^{-1}(z_{n+1}^{\\gamma,R} - z_n^{\\gamma,R}) - \\bE(\\gamma^{-1}(z_{n+1}^{\\gamma,R} - z_n^{\\gamma,R})|\\cF_n)\\,.\n$$\n\\begin{lemma}\n  \\label{lem:tightness-in-C}\nLet Assumptions~\\ref{hyp:model}, \\ref{hyp:alpha-beta}, \\ref{hyp:moment-f} and \\ref{hyp:iid} hold true.\nThere exists $\\bar \\gamma_0>0$ s.t. for every $R>0$, the family of r.v. $(\\sz^{\\gamma,R}:\\gamma\\in (0,\\bar \\gamma_0])$\nis tight. Moreover, for every $\\delta>0$,\n$ \n\\bP\\left(\\max_{0\\leq n\\leq \\lfloor\\frac T\\gamma\\rfloor}\\gamma\\left\\|\\sum_{k=0}^n\\Delta_{k+1}^{\\gamma,R}\\right\\|>\\delta\\right)\\xrightarrow[]{\\gamma\\to 0} 0\\,.\n\n\\end{lemma}\n\\begin{proof}\n  It is an immediate consequence of Lemma~\\ref{lem:UI} and \\cite[Lemma 6.2]{bianchi2019constant}\n\\end{proof}\n\\noindent The proof of the following lemma is omitted.\n\\begin{lemma}\n  \\label{lem:cv-h}\nLet Assumptions~\\ref{hyp:model} and \\ref{hyp:alpha-beta} hold true.\nConsider $t>0$ and $z\\in \\cZ_+$. Let $(\\varphi_n,z_n)$ be a sequence on $\\bN^*\\times \\cZ_+$ s.t.\n$\\lim_{n\\to\\infty}\\gamma_n\\varphi_n= t$ and $\\lim_{n\\to\\infty}z_n= z$. Then, $\\lim_{n\\to\\infty} h_{\\gamma_n}(\\varphi_n,z_n)= h(t,z)$\nand $\\lim_{n\\to\\infty} e_{\\gamma_n}(\\varphi_n,z_n)= \\bar e(t,z)$.\n\\end{lemma}\n\n\\noindent\\textbf{End of the Proof of Th.~\\ref{th:weak-cv}}\nConsider $x_0\\in \\bR^d$ and set $z_0=(x_0,0,0)$. Define\n$R_0\\eqdef \\sup\\left\\{\\|\\bar e(t,Z_\\infty^0(z_0)(t))\\|:t>0\\right\\}\\,.$\nBy Prop.~\\ref{prop:adam-bounded}, $R_0<+\\infty$. We select an arbitrary $R$ s.t. $R\\geq  R_0+1$.\nFor every $n\\geq 0$, $z\\in \\cZ_+$,\n$$\nz_{n+1}^{\\gamma,R} = z_{n}^{\\gamma,R}+\\gamma H_\\gamma(n+1,z_{n}^{\\gamma,R},\\xi_{n+1}) \\1_{\\|e_\\gamma(n,z_{n}^{\\gamma,R})\\|\\leq R}  \\,.\n$$\nDefine for every $n\\geq 1$, $z\\in \\cZ_+$,\n$h_{\\gamma,R}(n,z)\\eqdef h_\\gamma(n,z)\\1_{\\|e_\\gamma(n-1,z)\\|\\leq R}$. Then,\\\\\n$\n\\Delta_{n+1}^{\\gamma,R} = \\gamma^{-1}(z_{n+1}^{\\gamma,R} - z_n^{\\gamma,R}) - h_{\\gamma,R}(n+1,z_{n}^{\\gamma,R})\\,.\n$\nDefine also for every $n\\geq 0$, \\\\\n\n  M_n^{\\gamma,R} \\eqdef \\sum_{k=1}^n\\Delta_k^{\\gamma,R} =\\gamma^{-1}(z_n^{\\gamma,R} - z_0) - \\sum_{k=0}^{n-1} h_{\\gamma,R}(k+1,z_{k}^{\\gamma,R})\\,.\n$\nConsider $t\\geq 0$ and set $n\\eqdef \\lfloor t\/\\gamma\\rfloor$.\nFor any $T>0$, it holds that :\n\\begin{equation*}\n  \\sup_{t\\in [0,T]} \\left\\| \\sz^{\\gamma,R}(t)-z_0 - \\int_{0}^{t}h_{\\gamma,R}(\\lfloor s\/\\gamma\\rfloor+1,\\sz^{\\gamma,R}(\\gamma\\lfloor s\/\\gamma\\rfloor))ds\\right\\|\\\\\n \n  \\leq \\max_{0\\leq n\\leq \\lfloor T\/\\gamma\\rfloor+1} \\gamma \\|M_n^{\\gamma,R}\\|\\,.\n\\end{equation*}\nBy Lemma~\\ref{lem:tightness-in-C},\n\\begin{equation}\n  \\label{eq:Mn-cv-proba-zero}\n \\bP\\left( \\sup_{t\\in [0,T]} \\left\\| \\sz^{\\gamma,R}(t)-z_0 - \\int_{0}^{t}h_{\\gamma,R}\\left(\\lfloor s\/\\gamma\\rfloor+1,\\sz^{\\gamma,R}(\\gamma\\lfloor s\/\\gamma\\rfloor)\\right)ds\\right\\|\n>\\delta\\right) \\xrightarrow[]{\\gamma\\to 0} 0\\,.\n\\end{equation}\nAs a second consequence of Lemma~\\ref{lem:tightness-in-C}, the family of r.v. $(\\sz^{\\gamma,R}:0<\\gamma\\leq \\bar \\gamma_0)$ is tight, where $\\bar \\gamma_0$\nis chosen as in Lemma~\\ref{lem:tightness-in-C} (it does not depend on $R$).\nBy Prokhorov's theorem, there exists a sequence $(\\gamma_k:k\\in \\bN)$ s.t. $\\gamma_k\\to 0$ and\ns.t. $(\\sz^{\\gamma_k,R}:k\\in \\bN)$ converges in distribution to some probability measure $\\nu$ on $C([0,+\\infty),\\cZ_+)$.\nBy Skorohod's representation theorem, there exists a r.v. $\\sz$ on some probability space $(\\Omega',\\cF',\\bP')$, with distribution $\\nu$,\nand a sequence of r.v. $(\\sz_{(k)}:k\\in \\bN)$ on that same probability space where for each $k \\in \\bN$, the r.v. $\\sz_{(k)}$ has the same distribution\nas the r.v. $\\sz^{\\gamma_k,R}$,\nand s.t. for every $\\omega\\in\\Omega'$, $\\sz_{(k)}(\\omega)$ converges\nto $\\sz(\\omega)$ uniformly on compact sets. Now select a fixed $T>0$. According to Eq.~(\\ref{eq:Mn-cv-proba-zero}), the sequence\n$$\n\\sup_{t\\in [0,T]} \\left\\| \\sz_{(k)}(t)-z_0 - \\int_{0}^{t}h_{\\gamma_{k},R}\\left(\\lfloor s\/\\gamma_{k}\\rfloor+1,\\sz_{(k)}(\\gamma_{k}\\lfloor s\/\\gamma_{k}\\rfloor)\\right)ds\\right\\|\\,,\n$$\nindexed by $k\\in \\bN$, converges in probability to zero as $k\\to\\infty$. One can therefore extract a further subsequence $\\sz_{(\\varphi_k)}$,\ns.t. the above sequence converges to zero almost surely. In particular, since $\\sz_{(k)}(t)\\to \\sz(t)$ for every $t$, we obtain that\n\\begin{equation}\n\\sz(t) = z_0 + \\lim_{k\\to\\infty} \\int_{0}^{t}h_{\\gamma_{\\varphi_k},R}\\left(\\lfloor s\/\\gamma_{\\varphi_k}\\rfloor+1,\\sz_{(\\varphi_k)}(\\gamma_{\\varphi_k}\\lfloor s\/\\gamma_{\\varphi_k}\\rfloor)\\right)ds\\quad (\\forall t\\in [0,T])\\,.\\label{eq:cv_z}\n\\end{equation}\nConsider $\\omega\\in \\Omega'$ s.t. the r.v. $\\sz$ satisfies (\\ref{eq:cv_z}) at point $\\omega$.\nFrom now on, we consider that $\\omega$ is fixed, and we handle $\\sz$ as an element of $C([0,+\\infty),\\cZ_+)$,\nand no longer as a random variable.\nDefine $\\tau\\eqdef \\inf\\{t\\in [0,T]: \\|\\bar e(t,\\sz(t))\\|>R_0+\\frac 12\\}$ if the latter set is non-empty,\nand $\\tau\\eqdef T$ otherwise.\nSince $\\sz(0)=z_0$ and $\\|z_0\\|<R_0$, it holds that $\\tau>0$ using the continuity of $\\sz$.\nChoose any $(s,t)$ s.t. $0<s<t<\\tau$.\nNote that $\\sz_{(k)}(\\gamma_{k}\\lfloor s\/\\gamma_{k}\\rfloor)\\to \\sz(s)$\nand $\\gamma_k(\\lfloor s\/\\gamma_{k}\\rfloor+1)\\to s$. Thus, by Lemma~\\ref{lem:cv-h},\n$h_{\\gamma_{k}}\\left(\\lfloor s\/\\gamma_{k}\\rfloor+1,\\sz_{(k)}(\\gamma_{k}\\lfloor s\/\\gamma_{k}\\rfloor)\\right)$\nconverges to $h(s,\\sz(s))$ and\n$e_{\\gamma_k}(\\lfloor s\/\\gamma_{k}\\rfloor,\\sz_{(k)}(\\gamma_{k}\\lfloor s\/\\gamma_{k}\\rfloor))$\nconverges to $\\bar e(s,\\sz(z))\\,.$\nSince $s<\\tau$,  $\\bar e(s,\\sz(z))\\leq R_0+\\frac 12$. As $R\\geq R_0+1$, there exists a certain\n$K(s)$ s.t. for every $k\\geq K(s)$,\n$\n\\1_{\\|e_{\\gamma_k}(\\lfloor s\/\\gamma_{k}\\rfloor,\\sz_{(k)}(\\gamma_{k}\\lfloor s\/\\gamma_{k}\\rfloor))\\|\\leq R} = 1\\,.\n$\nAs a consequence, $h_{\\gamma_{k},R}(\\lfloor s\/\\gamma_{k}\\rfloor+1,\\sz_{(k)}(\\gamma_{k}\\lfloor s\/\\gamma_{k}\\rfloor))$\nconverges to $h(s,\\sz(s))$ as $k\\to\\infty$.\nUsing Lebesgue's dominated convergence theorem,  we obtain, for all $t\\in [0,\\tau]$:\n$\n\\sz(t) = z_0 +  \\int_{0}^{t}h\\left(s,\\sz(s))\\right)ds\\,\n$\nTherefore $\\sz(t) = Z_\\infty^0(x_0)(t)$ for every $t\\in [0,\\tau]$. In particular,\n$\\|\\sz(\\tau)\\|\\leq R_0$\nand this means that\n$\\tau = T$. Thus, $\\sz(t)=Z_\\infty^0(x_0)(t)$ for every $t\\in [0,T]$\n(and consequently for every $t\\geq 0$). We have shown that for every $R\\geq R_0+1$, the sequence of r.v.\n$(\\sz^{\\gamma,R}:\\gamma \\in (0,\\bar \\gamma_0])$ is tight and converges in probability to $Z_\\infty^0(z_0)$ as $\\gamma\\to 0$.\nTherefore, for every $T>0$,\n\\begin{equation}\n\\forall \\delta>0,\\ \\lim_{\\gamma\\to 0} \\bP\\left(\\sup_{t\\in [0,T]}\\left\\| \\sz^{\\gamma,R}(t) - Z_\\infty^0(x_0)(t)\\right\\|>\\delta \\right) =0\\,.\n\\label{eq:cv-proba-R}\n\\end{equation}\nIn order to complete the proof, we show that\n$\n\\bP\\left(\\sup_{t\\in [0,T]}\\left\\| \\sz^{\\gamma,R}(t) - \\sz^{\\gamma}(t)\\right\\|>\\delta \\right) \\to 0\n$\nas $\\gamma \\to 0$, for all $\\delta >0$.\nwhere we recall that $\\sz^\\gamma=\\sX_\\gamma(z^\\gamma)$.\nNote that $\\left\\| \\sz^{\\gamma,R}(t)\\right\\|\\leq\n\\left\\| \\sz^{\\gamma,R}(t)-Z_\\infty^0(z_0)(t)\\right\\| + R_0$ by the triangular inequality.\nTherefore, for every $T,\\delta>0$,\n\\begin{align*\n  \\bP\\left(\\sup_{t\\in [0,T]}\\left\\| \\sz^{\\gamma,R}(t) - \\sz^{\\gamma}(t)\\right\\|>\\delta \\right)\n  &\\leq \\bP\\left(\\sup_{t\\in [0,T]}\\left\\| \\sz^{\\gamma,R}(t)\\right\\|\\geq R \\right)\\\\\n  &\\leq \\bP\\left(\\sup_{t\\in [0,T]}\\left\\| \\sz^{\\gamma,R}(t)-Z_\\infty^0(z_0)(t)\\right\\| \\geq R-R_0\\right)\\,.\n\\end{align*\nBy Eq.~(\\ref{eq:cv-proba-R}), the RHS of the above inequality tends to zero as $\\gamma\\to 0$.\nThe proof is complete.\n\n\\subsection{Proof of Th.~\\ref{th:longrun}}\n\\label{sec:longrun}\n\n\nWe start by stating a general result. Consider a Euclidean space $\\sX$ equipped with its Borel $\\sigma$-field $\\cal X$.\nLet $\\bar \\gamma_0>0$, and consider two families\n$(P_{\\gamma,n}:0<\\gamma<\\bar \\gamma_0, n\\in \\bN^*)$ and $(\\bar P_{\\gamma}:0<\\gamma<\\bar \\gamma_0)$\nof Markov transition kernels on $\\sX$.\nDenote by $\\cP(\\sX)$ the set of probability measures on $\\sX$.\nLet $X=(X_n:n\\in\\bN)$ be the canonical process on $\\sX$.\nLet $(\\bP^{\\gamma,\\nu}:0<\\gamma<\\bar \\gamma_0,\\nu\\in \\cP(\\sX))$ and\n$(\\bar \\bP^{\\gamma,\\nu}:0<\\gamma<\\bar \\gamma_0,\\nu\\in \\cP(\\sX))$ be two families of measures on the canonical space\n$(X^\\bN,\\cal X^{\\otimes\\bN})$ such that the following holds:\n\\begin{itemize}[leftmargin=*]\n\\item Under $\\bP^{\\gamma,\\nu}$, $X$ is a non-homogeneous Markov chain with transition kernels $(P_{\\gamma,n}:n\\in \\bN^*)$\nand initial distribution $\\nu$, that is, for each $n\\in\\bN^*$,\n$\n\\bP^{\\gamma,\\nu}(X_{n}\\in dx|X_{n-1}) = P_{\\gamma,n}(X_{n-1},dx)\\,.\n$\n\\item Under $\\bar\\bP^{\\gamma,\\nu}$, $X$ is an homogeneous Markov chain with transition kernel $\\bar P_{\\gamma}$\nand initial distribution $\\nu$.\n\\end{itemize}\nIn the sequel, we will use the notation $\\bar P^{\\gamma,x}$ as a shorthand notation for\n$\\bar P^{\\gamma,\\delta_x}$ where $\\delta_x$ is the Dirac measure at some point $x\\in \\sX$.\nFinally, let $\\Psi$ be a semiflow on $\\sX$. A Markov kernel $P$ is \\emph{Feller}\nif $Pf$ is continuous for every bounded continuous  $f$.\n\\begin{assumption} Let $\\nu\\in \\cP(\\sX)$.\n  \\begin{enumerate}[{\\sl i)}]\n  \\item For every $\\gamma$, $\\bar P_{\\gamma}$ is Feller.\n  \\item $(\\bP^{\\gamma,\\nu}X_n^{-1}:n\\in \\bN,0<\\gamma<\\bar \\gamma_0)$ is a tight family of measures.\n  \\item For every $\\gamma\\in (0,\\bar \\gamma_0)$ and every bounded Lipschitz continuous function $f:\\sX\\to\\bR$,\n$P_{\\gamma,n}f$ converges to $\\bar P_\\gamma f$ as $n\\to\\infty$, uniformly on compact sets.\n\\item For every $\\delta>0$, for every compact set $K\\subset \\sX$, for every $t>0$,\\\\\n\n\\lim_{\\gamma\\to 0}\\sup_{x\\in K}\\bar P^{\\gamma,x}\\left(\\|X_{\\lfloor t\/\\gamma\\rfloor} - \\Psi_t(x)\\|>\\delta\\right)=0\\,.\n\n  \\end{enumerate}\n\\label{hyp:general}\n\\end{assumption}\nLet $BC_\\Psi$ be the Birkhof center of $\\Psi$ \\emph{i.e.}, the closure of the set of recurrent points.\n\n\\begin{theorem}\n\\label{longrun}\nConsider $\\nu\\in \\cP(\\sX)$ s.t. Assumption~\\ref{hyp:general} holds true. Then, for every $\\delta>0$,\n$\n\\lim_{\\gamma\\to 0}\\limsup_{n\\to\\infty} \\frac 1{n+1}\\sum_{k=0}^n \\bP^{\\gamma,\\nu}\\left(d(X_k,BC_\\Psi)>\\delta\\right)=0\\,.\n$\n\\end{theorem}\nWe omit the proof of this result which follows a similar reasoning to \\cite[Th.~5.5 and Proof in section 8.4]{bianchi2019constant} and makes use of results from \\cite{for-pag-99}.\n\n\\noindent\\textbf{End of the Proof of Th.~\\ref{th:longrun}.}\nWe apply Th.~\\ref{longrun} in the case where $P_{\\gamma,n}$ is the kernel\nof the non-homogeneous Markov chain $(z_n^\\gamma)$ defined by~(\\ref{eq:znT}) and\n$\\bar P_\\gamma$ is the kernel of the homogeneous Markov chain $(\\bar z_n^\\gamma)$\ngiven by\n$\\bar z_n^\\gamma = \\bar z_{n-1}^\\gamma+\\gamma H_\\gamma(\\infty,\\bar z_{n-1}^\\gamma,\\xi_n)$\nfor every $n\\in\\bN^*$ and $\\bar z_0 \\in \\cZ_+$ where $H_\\gamma(\\infty,\\bar z_{n-1}^\\gamma,\\xi_n) \\eqdef \\lim_{k \\to \\infty} H_\\gamma(k,\\bar z_{n-1}^\\gamma,\\xi_n)$. The task is merely to verify Assumption~\\ref{hyp:general}{\\sl iii)}, the other assumptions being easily verifiable using Th.~\\ref{th:weak-cv},\nConsider $\\gamma\\in (0,\\bar \\gamma_0)$. Let $f: \\cZ \\to \\mathbb{R}$ be a bounded $M$-Lipschitz continuous function and $K$ a compact.\nFor all $z=(x,m,v) \\in K$:\n\\begin{align*}\n&|P_{\\gamma,n}(f)(z) - \\bar P_{\\gamma}(f)(z)| \\leq M \\gamma \\bE \\left \\| \\frac{(1-\\alpha^n)^{-1}\\tilde{m}_\\xi}{ \\varepsilon+(1-\\beta^n)^{-\\frac{1}{2}}{\\tilde{v}_\\xi^{1\/2}}} -\\frac{\\tilde{m}_\\xi}{ \\varepsilon+{\\tilde{v}_\\xi^{1\/2}}}\\right \\| \\\\\n&\\resizebox{.99\\hsize}{!}{$\\leq \\frac{M \\gamma\\alpha^n}{\\varepsilon(1- \\alpha^n)} \\sup_{x,m}\\left(\\alpha ||m|| + (1-\\alpha)\\bE||\\nabla f(x,\\xi)|| \\right)\n + \\frac{M \\gamma \\bE ||\\tilde{m}_\\xi \\odot \\tilde{v}_\\xi^{1\/2}||}{\\varepsilon^2}\\left(1- \\frac{1}{(1-\\beta^n)^{1\/2}}\\right)$}\\,\n\\end{align*}\nwhere we write $\\alpha=\\bar \\alpha(\\gamma)$, $\\beta=\\bar \\beta(\\gamma)$,\n$\\tilde{m}_\\xi \\eqdef \\alpha m+(1-\\alpha)\\nabla f(x,\\xi)$ and\n$\\tilde{v}_\\xi \\eqdef  \\beta v+(1- \\beta)\\nabla f(x,\\xi)^{\\odot 2}$.\nThus, condition~\\ref{hyp:general}{\\sl iii)} follows.\nFinally, Cor.~\\ref{coro:cv} implies $BC_\\Phi=\\cE$.\n\n\\section{Proofs of Section~\\ref{sec:discrete_decreasing}}\n\\label{sec:proofs_sec_discrete_decreasing}\nIn this section, we denote by $\\bE_n = \\bE(\\cdot|\\cF_n)$ the conditional expectation w.r.t. $\\cF_n$.\nWe also use the notation $\\nabla f_{n+1} \\eqdef \\nabla f(x_n,\\xi_{n+1})$.\n\nThe following lemma will be useful in the proofs.\n\\begin{lemma}\n\\label{lemma:r_n}\nLet the sequence $(r_n)$ be defined as in Algorithm~\\ref{alg:adam-decreasing}.\nAssume that $0 \\leq \\alpha_n \\leq 1$ for all $n$ and that $(1-\\alpha_n)\/\\gamma_n \\to a > 0$ as $n \\to +\\infty$.\nThen,\n\\begin{enumerate}[{\\sl i)}]\n\\item $\\forall n \\in \\bN, r_n = 1 - \\prod_{i=1}^n \\alpha_i$,\n\\item The sequence $(r_n)$ is nondecreasing and converges to $1$.\n\\item Under Assumption~\\ref{hyp:step-tcl}~\\ref{step-tcl-i}, for every $\\epsilon>0$, for sufficiently large $n$, we have\n$r_n-1 \\leq e^{-\\frac{a\\gamma_0}{2(1-\\kappa)}n^{1-\\kappa}}$ if $\\kappa \\in (0,1)$ and\n$r_n-1 \\leq n^{-a \\gamma_0\/(1+\\epsilon)}$ if $\\kappa = 1$.\n\\end{enumerate}\n\\end{lemma}\nA similar lemma holds for the sequence $(\\bar{r}_n)$.\n\\begin{proof}\ni) stems from observing that $r_{n+1} -1 = \\alpha_{n+1}(r_n -1)$ for every $n \\in \\bN$ and\niterating this relation ($r_0=0$). As a consequence, the sequence $(r_n)$ is nondecreasing.\nWe can write :\n$\n0 \\leq 1-r_n \\leq \\exp(- \\sum_{i=1}^n (1-\\alpha_i))\\,.\n$\niii) As $\\sum_{n\\geq 1} \\gamma_n = + \\infty$\nand $(1-\\alpha_n) \\sim a\\gamma_n$, we deduce that $\\sum_{i=1}^n (1-\\alpha_i) \\sim \\sum_{i=1}^n a\\gamma_i$.\nThe results follow from the fact that $\\sum_{i=1}^n \\gamma_i \\sim \\frac{\\gamma_0}{1-\\kappa} n^{1-\\kappa}$\nwhen $\\kappa \\in (0,1)$ and $\\sum_{i=1}^n \\gamma_i \\sim \\gamma_0 \\ln n$ for $\\kappa=1$.\n\\end{proof}\n\n\\subsection{Proof of Th.~\\ref{thm:as_conv_under_stab}}\n\nWe define\n$\\bar z_n = (x_{n-1},m_n,v_n)$ (note the shift in the index of the variable $x$).\nWe have\n$$\n\\bar z_{n+1} = \\bar z_n + \\gamma_{n+1} h_\\infty(\\bar z_n) + \\gamma_{n+1}\\chi_{n+1} + \\gamma_{n+1} \\varsigma_{n+1}\\,,\n$$\nwhere $h_\\infty$ is defined in Eq.~(\\ref{eq:h_infty}) and where we set\n$$\n\\chi_{n+1} = \\left(0,\\gamma_{n+1}^{-1}(1-\\alpha_{n+1})(\\nabla f_{n+1}-\\nabla F(x_n)),\\gamma_{n+1}^{-1}(1-\\beta_{n+1})(\\nabla f_{n+1}^{\\odot 2}-S(x_n))\\right)\n$$\nand $\\varsigma_{n+1} = (\\varsigma_{n+1}^x,\\varsigma_{n+1}^m,\\varsigma_{n+1}^v)$ with the components defined by:\n$\\varsigma_{n+1}^x = \\frac{m_n}{\\varepsilon + \\sqrt{v_n}} - \\frac{\\gamma_n}{\\gamma_{n+1}}\\frac{\\hat m_n}{\\varepsilon + \\sqrt{\\hat v_n}}$,\n$\\varsigma_{n+1}^m = \\left(\\frac{1-\\alpha_{n+1}}{\\gamma_{n+1}}-a\\right)(\\nabla F(x_n)-m_n) + a(\\nabla F(x_n) - \\nabla F(x_{n-1}))$ and\n$\\varsigma_{n+1}^v = \\left(\\frac{1-\\beta_{n+1}}{\\gamma_{n+1}}-b\\right)(S(x_n)-v_n) + b(S(x_n) - S(x_{n-1}))$ \\,.\nWe prove that $\\varsigma_n\\to 0$ a.s.\nUsing the triangular inequality,\n\\begin{align*}\n  \\|\\varsigma_n^x\\|\n&\\leq \\left\\| \\frac{m_n}{\\varepsilon + \\sqrt{v_n}} -\\frac{m_n}{\\bar r_n^{1\/2}\\varepsilon + \\sqrt{v_n}}\\right\\|\n+ \\left|1-\\frac{\\gamma_nr_n^{-1}}{\\gamma_{n+1}\\bar r_n^{-1\/2}}\\right|\\left\\| \\frac{m_n}{\\bar r_n^{1\/2}\\varepsilon + \\sqrt{v_n}}\\right\\|\\\\\n&\\leq \\varepsilon^{-1}|1-\\bar r_n^{-1\/2}|\\|m_n\\| + \\varepsilon^{-1}\\left|\\bar r_n^{-1\/2}-\\frac{\\gamma_nr_n^{-1}}{\\gamma_{n+1}}\\right|\\|m_n\\|\\,,\n\\end{align*}\nwhich converges a.s. to zero because of the boundedness of $(z_n)$ combined with Assumption~\\ref{hyp:stepsizes} and Lemma~\\ref{lemma:r_n} for $(\\bar r_n)$.\nThe components $\\varsigma_{n+1}^m$ and $\\varsigma_{n+1}^v$ converge a.s. to zero,\nas products of a bounded term and a term converging to zero.\nIndeed, note that $\\nabla F$ and $S$ are locally Lipschitz continuous under Assumption~\\ref{hyp:model}.\nHence, there exists a constant $C$ s.t. $\\|\\nabla F(x_n) - \\nabla F(x_{n-1})\\| \\leq C \\|x_n-x_{n-1}\\| \\leq \\frac{C}{\\varepsilon}\\gamma_n\\|m_n\\|$.\nThe same inequality holds when replacing $\\nabla F$ by $S$.\nNow consider the martingale increment sequence $(\\chi_n)$, adapted to $\\cF_n$.\nEstimating the second order moments, it is easy to show using Assumption~\\ref{hyp:moment-f}~\\ref{momentegal} that\nthere exists a constant $C'$ s.t.\n$\\bE_n(\\|\\chi_{n+1}\\|^2)\\leq C'$.\nUsing that $\\sum_k\\gamma_k^2<\\infty$, it follows that $\\sum_n\\bE_n(\\|\\gamma_{n+1}\\chi_{n+1}\\|^2)<\\infty$ a.s.\nBy Doob's convergence theorem, $\\lim_{n\\to\\infty} \\sum_{k\\leq n} \\gamma_k\\chi_k$ exists almost surely.\nUsing this result along with the fact that $\\varsigma_n$ converges a.s. to zero, it follows from\nusual stochastic approximation arguments \\cite{ben-(cours)99} that the interpolated process\n$\\bar\\sz: [0,+\\infty)\\to \\cZ_+$ given by\n\\[\n\\bar \\sz(t) = \\bar z_n + (t-\\tau_n) \\frac{\\bar z_{n+1}-\\bar z_n}{\\gamma_{n+1}} \\qquad \\left(\\forall n \\in \\bN\\,,\\, \\forall t \\in [\\tau_n,\\tau_{n+1})\\right)\n\\]\n(where $\\tau_n = \\sum_{k=0}^n\\gamma_k$), is almost surely a bounded APT of the semiflow $\\bar\\Phi$ defined by~(\\ref{eq:ode-a}).\nThe proof is concluded by applying Prop.~\\ref{prop:benaim} and Prop.~\\ref{prop:Wstrict}.\n\n\\subsection{Proof of Prop.~\\ref{thm:stab}}\n\\label{sec:stability}\n\nAs $\\inf F>-\\infty$, one can assume without loss of generality that $F\\geq 0$.\nIn the sequel,\n$C$ denotes some positive constant which may change from line to line.\nWe define $a_n \\eqdef (1-\\alpha_{n+1})\/\\gamma_n$ and\n$P_n\\eqdef \\frac 1{2a_nr_n}\\ps{m_n^{\\odot 2},\\frac 1{\\varepsilon+\\sqrt{\\hat v_n}}}$.\nWe have $a_n\\to a$ and $r_n\\to 1$.\nBy Assumption~\\ref{hyp:stab}-\\ref{lipschitz},\n\\begin{align}\nF(x_{n}) \n&\\leq F(x_{n-1}) - \\gamma_{n} \\ps{\\nabla F(x_{n}),\\frac{\\hat m_n}{\\varepsilon + \\sqrt{\\hat v_n}}} + C\\gamma_n^2 P_n\\label{eq:lip}.\n\\end{align}\nWe set $u_n \\eqdef 1-\\frac{a_{n+1}}{a_{n}}$ and\n$D_n \\eqdef \\frac {r_n^{-1}}{\\varepsilon+\\sqrt{\\hat v_{n}}}$, so that $P_n = \\frac 1{2a_n}\\ps{D_n,m_n^{\\odot 2}}$.\nWe can write:\n\\begin{equation}\nP_{n+1}-P_n= u_nP_{n+1} +\\ps{\\frac{D_{n+1}-D_n}{2a_n},m_{n+1}^{\\odot 2}}+\\ps{\\frac{D_n}{2a_n},m_{n+1}^{\\odot 2}-m_n^{\\odot 2}}.\\label{eq:P-P}\n\\end{equation}\nWe estimate the vector $D_{n+1}-D_n$.\nUsing that $(r_n^{-1})$ is non-increasing,\n$$\nD_{n+1}-D_n \\leq r_{n}^{-1} \\frac{\\sqrt{\\hat v_n} - \\sqrt{\\hat v_{n+1}}}{(\\varepsilon + \\sqrt{\\hat v_{n+1}})\\odot(\\varepsilon + \\sqrt{\\hat v_n})}\\,.\n$$\nRemarking that $v_{n+1} \\geq \\beta_{n+1} v_n$, recalling that\n$(\\bar{r}_n)$ is nondecreasing and using the update rules of $v_n$ and\n$\\bar{r}_n$, we obtain after some algebra\n\\begin{align}\n  \\label{eq:subsubterm2}\n\\sqrt{\\hat v_n} - \\sqrt{\\hat v_{n+1}} &=  \\bar{r}_{n+1}^{-\\frac 12}(1-\\beta_{n+1})\\frac{v_n-\\nabla f_{n+1}^{\\odot 2}}{\\sqrt{v_n}+\\sqrt{v_{n+1}}\n                                          + \\frac{\\bar{r}_{n+1}-\\bar{r}_n}{\\sqrt{\\bar{r}_n}(\\sqrt{\\bar{r}_n}+\\sqrt{\\bar{r}_{n+1}})} \\sqrt{\\frac{v_n}{\\bar{r}_{n+1}}} \\nonumber\\\\\n                                         \n                                          &\\leq c_{n+1} \\sqrt{\\hat v_{n+1}}  \\,\\, \\text{where} \\,\\,\n                                          c_{n+1} \\eqdef \\frac{1-\\beta_{n+1}}{\\sqrt{\\beta_{n+1} }}\\left( \\frac{1}{1+\\sqrt{\\beta_{n+1}}} + \\frac{1- \\bar{r}_n}{2\\bar{r}_n} \\right)\\,.\n\\end{align}\nIt is easy to see that $c_n\/\\gamma_n\\to b\/2$. Thus, for any $\\delta>0$, $c_{n+1}\\leq (b+2\\delta)\\gamma_{n}\/2$ for all $n$ large enough.\nUsing also that $\\sqrt{\\hat v_{n+1}}\/(\\varepsilon +\\sqrt{\\hat v_{n+1}})\\leq 1$, we obtain that\n\nD_{n+1}-D_n \\leq\n\\frac{b+2\\delta}2 \\gamma_n D_n\\,.\n$\nSubstituting this inequality in Eq.~(\\ref{eq:P-P}), we get\n\\begin{align*}\n  P_{n+1}-P_n&\\leq u_nP_{n+1} +\\gamma_n\\ps{\\frac{b+2\\delta}{4a_n} D_n,m_{n+1}^{\\odot 2}}+\\ps{\\frac{D_n}{2a_n},m_{n+1}^{\\odot 2}-m_n^{\\odot 2}}\\,.\n\\end{align*}\nUsing $m_{n+1}^{\\odot 2} - m_n^{\\odot 2} =  2 m_n\n\\odot (m_{n+1}-m_n)+(m_{n+1}-m_n)^{\\odot 2} $, and noting that\\\\ $\\bE_n(m_{n+1}-m_n) = a_n\\gamma_n(\\nabla F(x_n) -m_n)$,\n\\begin{equation*}\n  \\bE_n\\ps{\\frac {D_n}{2a_n},m_{n+1}^{\\odot 2}-m_n^{\\odot 2}}\n = \\gamma_n\\ps{\\nabla F(x_n),\\frac{\\hat m_n}{\\varepsilon+\\sqrt{\\hat v_n}}}-2a_n\\gamma_nP_n+\\ps{\\frac {D_n}{2a_n}, \\bE_n[(m_{n+1}-m_n)^{\\odot 2}] }\n\\end{equation*}\nAs $a_n\\to a$, we have $a_n-\\frac{b+2\\delta}{4}\\geq a-\\frac{b+\\delta}{4}$ for all $n$ large enough. Hence,\n\\begin{align*}\n \\bE_n P_{n+1}-P_n&\\leq\nu_nP_{n+1} -2(a-\\frac{b+\\delta}{4})\\gamma_n P_n + \\gamma_n\\ps{\\nabla F(x_n),\\frac{\\hat m_n}{\\varepsilon+\\sqrt{\\hat v_n}}}\n\\\\\n&+\n\\gamma_n^2\\frac{b+2\\delta}2\\ps{\\nabla F(x_n),\\frac{\\hat m_n}{\\varepsilon+\\sqrt{\\hat v_n}}}\n+C\\ps{\\frac {D_n}{2a_n}, \\bE_n[(m_{n+1}-m_n)^{\\odot 2}] }\\,.\n\\end{align*}\nUsing the Cauchy-Schwartz inequality and Assumption~\\ref{hyp:stab}~\\ref{momentgrowth},\nit is easy to show the inequality\n$\\ps{\\nabla F(x_n),\\frac{\\hat m_n}{\\varepsilon+\\sqrt{\\hat v_n}}}\\leq C(1+F(x_n)+P_n)$.\nMoreover, using the componentwise inequality $(\\nabla f_{n+1}-m_n)^{\\odot 2} \\leq 2 \\nabla f_{n+1}^{\\odot 2} + 2 m_n^{\\odot 2}$\nalong with Assumption~\\ref{hyp:stab}~\\ref{momentgrowth}, we obtain\n\\begin{equation*}\n  \\resizebox{\\hsize}{!}{$\n  \\ps{\\frac {D_n}{2a_n}, \\bE_n[(m_{n+1}-m_n)^{\\odot 2}] } \\leq\n2(1-\\alpha_{n+1})^2\\ps{\\frac {D_n}{2a_n},\n\\bE_n[ \\nabla f_{n+1}^{\\odot 2}]+ m_n^{\\odot 2}\n}\n\\leq C\\gamma_n^2(1+F(x_n)+P_n)\\,.\n$}\n\\end{equation*}\nPutting all pieces together with Eq.~(\\ref{eq:lip}),\n\\begin{equation}\n\\label{eq:F+P}\n\\resizebox{0.95\\hsize}{!}{$\n \\bE_n(F(x_n)+ P_{n+1}) \\leq F(x_{n-1})+P_{n}\n +u_nP_{n+1} -2(a-\\frac{b+\\delta}{4})\\gamma_n P_n\n+C\\gamma_n^2(1+F(x_n)+P_n)\\,.\n$}\n\\end{equation}\nDefine\n$\nV_n\\eqdef (1-C\\gamma_{n-1}^2)F(x_{n-1})+(1-u_{n-1})P_{n}\n$\nwhere the constant $C$ is fixed so that\nEq.~(\\ref{eq:F+P}) holds.\nThen,\n\\begin{equation*}\n  \\bE_n(V_{n+1}) \\leq V_n\n -\\left(2a-\\frac{b+\\delta}{2}-\\frac{u_{n-1}}{\\gamma_n}\\right)\\gamma_n P_n\n+C\\gamma_n^2(1+P_n)+ C\\gamma_{n-1}^2F(x_{n-1})\\,.\n\\end{equation*}\nBy Assumption~\\ref{hyp:stab}, $\\lim\\sup_n u_{n-1}\/\\gamma_n< 2a-b\/2$ and for $\\delta$ small enough, we obtain\n\\begin{equation*}\n  \\bE_n(V_{n+1}) \\leq V_n\n+C\\gamma_n^2(1+P_n)+ C\\gamma_{n-1}^2F(x_{n-1})\\leq (1+ C'\\gamma_n^2)V_n +C\\gamma_n^2\\,.\n\\end{equation*}\nBy the Robbins-Siegmund's theorem  \\cite{robbins1971convergence},\nthe sequence $(V_n)$ converges almost surely to a finite random variable $V_\\infty \\in \\bR^+$.\nIn turn, the coercivity of $F$ implies that $(x_n)$ is almost surely bounded.\nWe now establish the almost sure boundedness of $(m_n)$.\nConsider the martingale difference sequence $\\Delta_{n+1}\\eqdef \\nabla f_{n+1} -\\nabla F(x_n)$.\nWe decompose $m_{n} = \\bar m_n + \\tilde m_n$ where\n$\\bar m_{n+1} = \\alpha_{n+1} \\bar m_n + (1-\\alpha_{n+1})\\nabla F(x_n)$ and\n$\\tilde m_{n+1} = \\alpha_{n+1} \\tilde m_n + (1-\\alpha_{n+1})\\Delta_{n+1}$, setting $\\bar m_0=\\tilde m_0=0$.\nWe prove that both terms $\\bar m_n$ and $\\tilde m_n$ are bounded. Consider the first term:\n$\n\\|\\bar m_{n+1}\\|\\leq \\alpha_{n+1} \\|\\bar m_n\\| + (1-\\alpha_{n+1}) \\sup_k\\|\\nabla F(x_k)\\|\\,.\n$\nBy continuity of $\\nabla F$, the supremum in the above inequality is almost surely finite.\nThus, for every $n$, the ratio $\\|\\bar m_n\\|\/\\sup_k\\|\\nabla F(x_k)\\|$ is upperbounded by the bounded sequence\n$r_n$. Hence, $(\\bar m_n)$ is bounded w.p.1. Consider now the term $\\tilde m_n$:\n\\begin{equation*}\n    \\resizebox{\\hsize}{!}{$\n  \\bE_n(\\|\\tilde m_{n+1}\\|^2)  = \\alpha_{n+1}^2\\|\\tilde m_n\\|^2 +  (1-\\alpha_{n+1})^2\\bE_n(\\|\\Delta_{n+1}\\|^2)\n                               \\leq  (1+(1-\\alpha_{n+1})^2)\\|\\tilde m_n\\|^2 +  (1-\\alpha_{n+1})^2C\\,,\n    $}\n\\end{equation*}\nwhere $C$ is a constant s.t. $\\bE_n(\\|\\nabla f_{n+1}\\|^2) \\leq C$ by Assumption~\\ref{hyp:moment-f}~\\ref{momentegal}.\nHere, we used $\\alpha_{n+1}^2\\leq (1+(1-\\alpha_{n+1})^2)$ and the inequality\n$\\bE_n(\\|\\Delta_{n+1}\\|^2) \\leq \\bE_n(\\|\\nabla f_{n+1}\\|^2)$.\nBy Assumption~\\ref{hyp:stepsizes}, $\\sum_n(1-\\alpha_{n+1})^2<\\infty$. By the Robbins-Siegmund theorem,\nit follows that $\\sup_n\\|\\tilde m_n\\|^2<\\infty$ w.p.1. Finally, it can be shown that $(v_n)$ is\nalmost surely bounded using the same arguments.\n\n\\subsection{Proof of Th.~\\ref{thm:clt}}\n\\label{sec:clt}\n\nWe use~\\cite[Th.~1]{pelletier1998weak}.\nAll the assumptions in the latter can be verified in our case, at the exception of\na positive definiteness condition on the\nlimiting covariance matrix, which corresponds, in our case, to the matrix $Q$\ngiven by Eq.~(\\ref{eq:Q}). As $Q$ is not positive definite, it is strictly speaking not possible\nto just cite and apply \\cite[Th.~1]{pelletier1998weak}. Nevertheless,\na detailed inspection of the proofs of \\cite{pelletier1998weak} shows that only a minor\nadaptation is needed in order to cover the present case.\nTherefore, proving the convergence result of \\cite{pelletier1998weak} from scratch is worthless.\nIt is sufficient to verify the assumptions of \\cite[Th.~1]{pelletier1998weak}\n(except the definiteness of $Q$) and then to point out the specific part of the proof of \\cite{pelletier1998weak}\nwhich requires some adaptation.\n\nLet $z_n=(x_n,m_n,v_n)$ be the output of Algorithm~\\ref{alg:adam-decreasing}.\nDefine $z^*=(x^*,0,S(x^*))$.\nDefine $\\eta_{n+1} \\eqdef (0, a(\\nabla f_{n+1} - \\nabla F(x_n)), b(\\nabla f_{n+1}^{\\odot 2}- S(x_n)))$.\nWe have\n\\begin{equation}\n  \\label{eq:zn_dec}\n  z_{n+1} = z_n + \\gamma_{n+1} h_\\infty(z_n) + \\gamma_{n+1}\\eta_{n+1} + \\gamma_{n+1} \\epsilon_{n+1}\\,,\n\\end{equation}\nwhere $\\epsilon_{n+1} \\eqdef (\\epsilon_{n+1}^1,\\epsilon_{n+1}^2,\\epsilon^3_{n+1})$, whose components are given by\n\\begin{equation*}\n  \\resizebox{\\hsize}{!}{$\n  \\epsilon_{n+1}^1 =  \\frac{m_n}{\\varepsilon+\\sqrt{v_n}} - \\frac{\\hat m_{n+1}}{\\varepsilon+\\sqrt{\\hat v_{n+1}}};\\,\n  \\epsilon_{n+1}^2 = \\left(\\frac{1-\\alpha_{n+1}}{\\gamma_{n+1}}-a\\right)\\left( \\nabla f_{n+1} - m_n\\right);\\,\n  \\epsilon_{n+1}^3 =  \\left(\\frac{1-\\beta_{n+1}}{\\gamma_{n+1}}-b\\right)\\left( \\nabla f_{n+1}^{\\odot 2} - v_n\\right)\\,.\n  $}\n\\end{equation*}\nHere, $\\eta_{n+1}$ is a martingale increment noise and\n$\\epsilon_{n+1} = (\\epsilon_{n+1}^1,\\epsilon_{n+1}^2,\\epsilon^3_{n+1})$ is a remainder term.\nThe aim is to check the assumptions (A1.1) to (A1.3) of\n\\cite{pelletier1998weak}, where the role of the quantities ($h$,\n$\\varepsilon_n$, $r_n$, $\\sigma_n$, $\\alpha$, $\\rho$, $\\beta$) in\n\\cite{pelletier1998weak} is respectively played by the quantities\n($h_\\infty$, $\\eta_n$, $\\epsilon_n$, $\\gamma_n$, $\\kappa$, $1$, $1$) of\nthe present paper.\n\nLet us first consider Assumption (A1.1) for $h_\\infty$.\nBy construction, $h_\\infty(z^*) = 0$. By Assumptions~\\ref{hyp:mean_field_tcl} and \\ref{hyp:S>0},\n$h_\\infty$ is continuously differentiable in the neighborhood of $z^*$ and its\nJacobian at $z^*$ coincides with the matrix $H$ given by Eq.~(\\ref{eq:H}).\nAs already discussed, after some algebra,\nit can be shown that the largest real part of the eigenvalues of $H$ coincides with $-L$ where $L>0$\nis given by Eq.~(\\ref{eq:L}).\nHence, Assumption (A1.1) of \\cite{pelletier1998weak}\nis satisfied for $h_\\infty$. Assumption (A1.3) is trivially satisfied\nusing Assumption~\\ref{hyp:step-tcl}. The crux is therefore to verify Assumption (A1.2).\nClearly, $\\bE(\\eta_{n+1}|\\cF_n)=0$. Using Assumption~\\ref{hyp:moment-f}\\ref{momentreinforce},\nit follows from straightforward manipulations based on Jensen's inequality\nthat for any $M>0$,\nthere exists $\\delta>0$ s.t.\n\n\\sup_{n\\geq 0}\\bE_n\\left(\\|\\eta_{n+1}\\|^{2+\\delta}\\right) \\1_{\\{\\|z_n-z^*\\|\\leq M\\}}<\\infty\\,.\n$\nNext, we verify the condition\n\\begin{equation}\n  \\label{eq:cond-reste}\n  \\lim_{n\\to\\infty} \\bE\\left(\\gamma_{n+1}^{-1}\\|\\epsilon_{n+1}\\|^2\\1_{\\{\\|z_n-z^*\\|\\leq M\\}}\\right) = 0\\,.\n\\end{equation}\nIt is sufficient to verify the latter for $\\epsilon^i_n$ ($i=1,2,3$) in place of $\\epsilon_n$.\nThe map $(m,v)\\mapsto m\/(\\varepsilon+\\sqrt{v})$ is Lipschitz continuous in a neighborhood of\n$(0,S(x^*))$ by Assumption~\\ref{hyp:S>0}. Thus, for $M$ small enough, there exists a constant $C$ s.t.\nif $ \\|z_n-z^*\\|\\leq M$, then\n\n  \\|\\epsilon_{n+1}^1\\| \\leq C\\left\\|r_{n+1}^{-1}m_{n+1}-m_n\\right\\|+C\\left\\|\\bar r_{n+1}^{-1} v_{n+1}-v_n\\right\\|\\,.\n$\nUsing the triangular inequality and the fact that $r_{n+1},\\bar r_{n+1}$ are bounded sequences away from zero, there exists an other constant $C$ s.t.\n\\begin{align*}\n  \\|\\epsilon_{n+1}^1\\| &\\leq C\\left\\|m_{n+1}-m_n\\right\\|+C\\left\\|v_{n+1}-v_{n}\\right\\|\n+C|r_{n+1}-1|+C|\\bar r_{n+1}-1|\\,.\n\\end{align*}\nUsing Lemma~\\ref{lemma:r_n} under Assumption~\\ref{hyp:step-tcl} (note that $\\gamma_0 > 1\/2L \\geq 1\/a$ when $\\kappa=1$),\nwe obtain that the sequence $|r_{n}-1|\/\\gamma_n$ is bounded, thus $|r_{n+1}-1|\\leq C\\gamma_{n+1}$.\n The sequence $(1-\\alpha_{n})\/\\gamma_n$ being also bounded, it holds that\n\\begin{align*}\n  \\|m_{n+1}-m_n\\|^2 \\1_{\\{\\|z_n-z^*\\|\\leq M\\}} \\leq  C\\gamma_{n+1}^2(1+\\|\\nabla f_{n+1}\\|^2 )\\1_{\\{\\|z_n-z^*\\|\\leq M\\}}\\,.\n\\end{align*}\nBy Assumption~\\ref{hyp:moment-f}~\\ref{momentreinforce},\n$\\bE_n(\\|\\nabla f_{n+1}\\|^2|)$ is bounded by a deterministic constant on $\\{\\|z_n-z^*\\|\\leq M\\}$.\nThus, $\\bE_n(\\|m_{n+1}-m_n\\|^2\\1_{\\{\\|z_n-z^*\\|\\leq M\\}})\\leq C\\gamma_{n+1}^2$.\nA similar result holds for $\\|v_{n+1}-v_n\\|^2$. We have thus shown that\n$ \\bE_n\\left(\\|\\epsilon_{n+1}^1\\|^2\\1_{\\{\\|z_n-z^*\\|\\leq M\\}}\\right)\\leq C\\gamma_{n+1}^2$. Hence,\nEq.~(\\ref{eq:cond-reste}) is proved for $\\epsilon_{n+1}^1$ in place of $\\epsilon_{n+1}$.\nUnder Assumption~\\ref{hyp:step-tcl}, the proof uses the same kind of arguments for $\\epsilon_{n+1}^2$, $\\epsilon_{n+1}^3$  and is omitted.\nFinally, Eq.~(\\ref{eq:cond-reste}) is proved.\nContinuing the verification of Assumption (A1.2), we establish that\n\\begin{equation}\n  \\label{eq:lim-cov}\n  \\bE_n(\\eta_{n+1}\\eta_{n+1}^T) \\to Q\\textrm{ a.s. on } \\{z_n\\to z^*\\}\\,.\n\\end{equation}\nDenote by $\\bar Q(x)$ the matrix given by the righthand side of Eq.~(\\ref{eq:Q}) when $x^*$\nis replaced by an arbitrary $x\\in \\cV$. It is easily checked that $\\bE_n(\\eta_{n+1}\\eta_{n+1}^T)=\\bar Q(x_n)$\nand by continuity, $\\bar Q(x_n)\\to Q$ a.s. on $\\{z_n\\to z^*\\}$, which proves (\\ref{eq:lim-cov}).\nTherefore, Assumption (A1.2) is fulfilled, except for the point mentioned at the beginning of this section :\n\\cite{pelletier1998weak} puts the additional condition that the limit\nmatrix in Eq.~(\\ref{eq:lim-cov}) is positive definite. This condition is not satisfied in our case,\nbut the proof can still be adapted. The specific part of the proof where the positive definiteness\ncomes into play is Th.~7 in \\cite{pelletier1998weak}. The proof of \\cite[Th.~1]{pelletier1998weak} can therefore\nbe adapted to the case of a positive semidefinite matrix.\nIn the proof of \\cite[Th.~7]{pelletier1998weak}, we only substitute the inverse of the square root of $Q$ by the Moore-Penrose inverse.\nFinally, the uniqueness of the stationary distribution $\\mu$ and its expression follow from \\cite[Th.~6.7, p. 357]{karatzas-shreve1991}.\n\n\\noindent\\textbf{Proof of Eq.~(\\ref{eq:cov})}.\nWe introduce the $d\\times d$ blocks of the $3d\\times 3d$ matrix\n$\\Sigma = \\left(\\Sigma_{i,j}\\right)_{i,j=1,2,3}$\nwhere $\\Sigma_{i,j}$ is $d\\times d$. We denote by $\\tilde\\Sigma$ the $2d\\times 2d$ submatrix $\\tilde\\Sigma \\eqdef\\left(\\Sigma_{i,j}\\right)_{i,j=1,2}$.\nBy Th.~\\ref{thm:clt}, we have the subsystem:\n\\begin{equation}\n\\tilde H \\tilde\\Sigma + \\tilde\\Sigma\\tilde H^T =\n\\begin{pmatrix}\n  0 & 0 \\\\\n0 & -a^2 \\tilde Q\n\\end{pmatrix}\\qquad\\text{where }\\tilde H \\eqdef\n\\begin{pmatrix}\n  \\zeta I_d & -D \\\\\na\\nabla^2F(x^*) & (\\zeta-a) I_d\n\\end{pmatrix}\\label{eq:lyap-reduced}\n\\end{equation}\nand where $\\tilde Q \\eqdef \\textrm{Cov}\\left(\\nabla\n  f(x^*,\\xi)\\right)$.  The next step is to triangularize the matrix\n$\\tilde H$ in order to decouple the blocks of $\\tilde\\Sigma$.  For\nevery $k=1,\\dots,d$, set\n$\\nu_k^\\pm \\eqdef -\\frac{a}{2}\\pm\\sqrt{a^2\/4-a\\lambda_k}$\nwith the convention that $\\sqrt{-1} =\n\\imath$ (inspecting the characteristic polynomial of $\\tilde H$, these\nare the eigenvalues of $\\tilde H$). Set\n$M^\\pm\\eqdef\\diag{(\\nu_1^\\pm,\n\\cdots,\\nu_d^\\pm)}$ and $R^\\pm\\eqdef\nD^{-1\/2}PM^\\pm P^TD^{-1\/2}$.  Using the identities $M^++M^-=-a I_d$ and\n$M^+M^-=a\\Lambda$ where\n$\\Lambda\\eqdef\\diag{(\\lambda_1,\\cdots,\\lambda_d)}$, it can be checked\nthat\n$$\n\\cR\\tilde H =\\begin{pmatrix}\n  D R^+ + \\zeta I_d & -D \\\\ 0 & R^-D + \\zeta I_d\n\\end{pmatrix}\\cR,\\text{ where }\\cR\\eqdef\n\\begin{pmatrix}\n  I_d & 0 \\\\ R^+ & I_d\n\\end{pmatrix}\\,.\n$$\nSet $\\check \\Sigma \\eqdef \\cR\\tilde\\Sigma\\cR^T$. Denote by\n$(\\check \\Sigma_{i,j})_{i,j=1,2}$ the blocks of $\\check\\Sigma$.\nNote that $\\check\\Sigma_{1,1} = \\Sigma_{1,1}$. By left\/right multiplication of Eq.~(\\ref{eq:lyap-reduced})\nrespectively with $\\cR$ and $\\cR^T$, we obtain\n\\begin{align}\n  &(DR^++\\zeta I_d) \\Sigma_{1,1}+\\Sigma_{1,1}(R^+D+\\zeta I_d) = \\check\\Sigma_{1,2} D+ D\\check\\Sigma_{1,2}^T \\label{eq:lapremiere}\\\\\n& (DR^++\\zeta I_d) \\check\\Sigma_{1,2}+\\check\\Sigma_{1,2} (DR^-+\\zeta I_d) = D\\check\\Sigma_{2,2} \\label{eq:ladeuxieme}\\\\\n& (R^-D+\\zeta I_d) \\check\\Sigma_{2,2} + \\check\\Sigma_{2,2} (DR^-+\\zeta I_d) = -a^2\\tilde Q \\label{eq:laderniere}\n\\end{align}\nSet $\\bar \\Sigma_{2,2} = P^{-1}D^{1\/2}\\check\\Sigma_{2,2} D^{1\/2}P$.\nDefine $C\\eqdef P^{-1}D^{1\/2}\\tilde Q D^{1\/2}P$.\nEq.~(\\ref{eq:laderniere}) yields $(M^-+\\zeta I_d) \\bar\\Sigma_{2,2} + \\bar\\Sigma_{2,2} (M^-+\\zeta I_d) = -a^2C$.\nSet $\\bar \\Sigma_{1,2} = P^{-1}D^{-1\/2}\\check\\Sigma_{1,2} D^{1\/2}P$.\nEq.~(\\ref{eq:ladeuxieme}) rewrites $(M^++\\zeta I_d) \\bar \\Sigma_{1,2}+\\bar \\Sigma_{1,2} (M^-+\\zeta I_d) = \\bar \\Sigma_{2,2}$.\nWe obtain that\n\n\\bar\\Sigma_{1,2}^{k,\\ell} = (\\nu_k^++\\nu_\\ell^-+2\\zeta)^{-1}\\bar\\Sigma_{2,2}^{k,\\ell} = \\frac{-a^2C_{k,\\ell}}{(\\nu_k^++\\nu_\\ell^-+2\\zeta)(\\nu_k^-+\\nu_\\ell^-+2\\zeta)}\\,.\n$\nSet\n$\\bar \\Sigma_{1,1} = P^{-1}D^{-1\/2}\\Sigma_{1,1} D^{-1\/2}P$.\nEq.~(\\ref{eq:lapremiere}) becomes $(M^++\\zeta I_d)\\bar \\Sigma_{1,1} + \\bar \\Sigma_{1,1}(M^++\\zeta I_d) = \\bar \\Sigma_{1,2} + \\bar \\Sigma_{1,2}^T$. Thus,\n\\begin{align*}\n\\bar\\Sigma_{1,1}^{k,\\ell} &\n\\resizebox{.9\\hsize}{!}{$\n= \\frac{\\bar\\Sigma_{1,2}^{k,\\ell}+\\bar\\Sigma_{1,2}^{\\ell,k}}{\\nu_k^++\\nu_\\ell^++2\\zeta}\n=  \\frac{-a^2C_{k,\\ell}}{(\\nu_k^++\\nu_\\ell^++2\\zeta)(\\nu_k^-+\\nu_\\ell^-+2\\zeta)}\\left(\\frac{1}{\\nu_k^++\\nu_\\ell^-+2\\zeta}\n+\\frac{1}{\\nu_k^-+\\nu_\\ell^++2\\zeta}\\right)\n$}\n\\\\ &\n= \\frac{C_{k,\\ell}}{ (1 - \\frac{2\\zeta}{a})(\\lambda_k+\\lambda_\\ell-2\\zeta + \\frac 2a \\zeta^2) +\\frac 1{2(a-2\\zeta)}(\\lambda_k-\\lambda_\\ell)^2}\\,,\n\\end{align*}\nand  the result is proved.\n\n\\bibliographystyle{siamplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}