\section{Introduction}\label{intro}

This work presents an extension of the hypocoercive approach studied in~\cite{brigati2022time} to the case of a non-zero potential energy function, and also considers the case of general kinetic energies beyond the standard quadratic one. Such generalizations can be relevant to enhance the performance of certain sampling methods in molecular dynamics and Bayesian statistics; see for instance~\cite{ST18,LFR19}. The more general framework also allows to make more transparent the structural assumptions behind the algebraic computations of~\cite{cao2019explicit,albritton2019variational}.

Kinetic models describe interacting multi-agent or multi-particle systems at an intermediate level between full microscopic and macroscopic scales \cite{balian2007microphysics}. More precisely, a kinetic partial differential equation (PDE) encodes the time evolution of the \textit{distribution function} of particles with respect to \textit{position} and some \textit{momentum} variable. Since the XIXth century, kinetic PDEs are formulated as a coupling between a transport and a collision/relaxation term. This is the case of the celebrated Boltzmann equation, for example. A general theory for Boltzmann's equation is still open. However, due to its physical validity, various simplified versions of it have been studied. A remarkable one is the Vlasov--Fokker--Planck equation, derived from Boltmann's equation in a limiting regime corresponding to grazing collisions \cite{degond1992fokker}. Vlasov--Fokker--Planck equations date back at least to \cite{kolmogoroff1934zufallige}, and they are useful models in statistical physics \cite{balian2007microphysics} and stellar dynamics~\cite{RevModPhys.15.1}. In general, Vlasov--Fokker--Planck equations should be understood as models for the time evolution of a distribution of particles subject to a potential and random background force inducing relaxation towards equilibrium. This paper is concerned with a rather general class of Vlasov--Fokker--Planck equations we introduce below.

We consider systems described by their configuration~$(x,v)$, where~$x \in \mathcal{X}$ is a spatial variable representing the positions of the particles composing the system, and~$v \in \mathcal{V}$ their velocities. Typical choices for the position space are $\mathcal{X} = \mathbb R^d$ for unconfined systems on the full space, or $\mathcal{X} = (L\mathbb{T})^d$ (with~$\mathbb{T} = \mathbb{R} \backslash \mathbb{Z}$ the unit torus) for systems enclosed in a cubic box of size~$L>0$ with periodic boundary conditions, a situation typical in molecular dynamics. The velocity space usually is~$\mathcal{V} = \mathbb{R}^d$, but here as well one could consider~$\mathcal{V} = \mathbb{T}^d$. In the latter case, and more generally when considering non-quadratic kinetic energies, one should rather think of~$v$ as being a momentum variable, the variable conjugated to the positions, rather than a velocity. The Hamiltonian associated with the system is supposed to be separable, and is therefore the sum~$\phi(x) + \psi(v)$ of the potential energy function~$\phi:\mathcal{X}\to \mathbb{R}$ and the kinetic energy~$\psi:\mathcal{V}\to\mathbb{R}$. 
We introduce the following measures:
\[
\mu(dx) := \mathrm{e}^{-\phi(x)} \, dx, \qquad \gamma(v) = \mathrm{e}^{-\psi(v)} \, dv.
\]
In order to have an invariant Boltzmann--Gibbs probability measure
\[
\Theta (dx \, dv) = \mu(dx) \, \gamma(dv).
\]
we shall assume that~$\mathrm{e}^{-\phi} \in L^1(\mathcal{X})$ and~$\mathrm{e}^{-\psi} \in L^1(\mathcal{V})$, and, without loss of generality, that $\mathrm{e}^{-\phi}$ and~$\mathrm{e}^{-\psi}$ integrate to~1, upon adding a constant to to~$\phi,\psi$. The Vlasov--Fokker--Planck equation we study reads
\begin{equation}
    \label{VFP}\tag{VFP}
    \left\{ \begin{aligned}
        & \partial_t f + \nabla_v \psi \cdot \nabla_x f - \nabla_x \phi \cdot \nabla_v f = \xi \, \nabla_v \cdot \left( \gamma \, \nabla_v \, \left( \frac{f}{\gamma} \right) \right), \\
        & f(t=0,\cdot,\cdot) = f_0 \in \mathrm{L}^2(\Theta^{-1}),
    \end{aligned} \right. 
\end{equation}
where $\xi > 0$ is a parameter regulating the intensity of the collision/relaxation term. 
We call 
$$Tf := \nabla_v \psi \cdot \nabla_x f  - \nabla_x \phi \cdot \nabla_v f, \qquad \mathscr Sf := \nabla_v \cdot \left( \gamma \, \nabla_v \, \left( \frac{f}{\gamma} \right) \right),$$
the transport and the collision operator, respectively. 

In order to highlight the role of $\xi,$ we briefly discuss a microscopic derivation of \eqref{VFP}, seen as the PDE governing the evolution of the law of Langevin dynamics. The Langevin dynamics associated with the Hamiltonian~$\phi(x)+\psi(v)$ read
\begin{equation}
  \label{eq:Langevin}
\left\{ \begin{aligned}
  dX_t & = \nabla_v \psi(V_t) \, dt, \\
  dV_t & = -\nabla_x \phi(X_t) \, dt - \xi \nabla_v \psi(V_t) \, dt + \sqrt{2\xi} \, dW_t,
\end{aligned} \right.
\end{equation}
where~$\xi > 0$ is the friction coefficient and~$(W_t)_{t \geq 0}$ is a standard $d$-dimensional Brownian motion. Let us motivate that, as for systems with quadratic kinetic energies, we expect the relaxation time of~\eqref{eq:Langevin} to the stationary state to be of order~$\max(\xi,\xi^{-1})$. In the limit~$\xi \to 0$, Langevin dynamics reduce to Hamiltonian dynamics. For small frictions, the average energy evolves on timescales of order~$\xi^{-1}$ since, by Itô calculus,
\[
\frac{d}{dt} \mathbb{E}\left[H(X_t,V_t)\right] = \mathbb{E}\left[(\mathcal{L}_\xi H)(X_t,V_t)\right] = \xi \mathbb{E}\left[(\mathcal{S} H)(X_t,V_t)\right],
\]
where~$\mathcal{L}_\xi = T + \xi \mathcal{S}$ is the generator of~\eqref{eq:Langevin}. The other limit of interest is the overdamped regime~$\xi \to +\infty$, where one recovers the overdamped Langevin dynamics upon rescaling time by a factor~$\xi$:
\[
d\overline{X}_t = -\nabla_x \phi(\overline{X}_t) \, dt + \sqrt{2} \, dB_t,
\]
where~$(B_t)_{t \geq 0}$ is also a standard $d$-dimensional Brownian motion. The formal proof of this limit relies on the observation that solutions of~\eqref{eq:Langevin} satisfy (upon replacing the term~$- \xi \nabla_v \psi (V_t) \, dt$ in the second line of~\eqref{eq:Langevin} by~$dX_t$ and integrating in time from~0 to~$\xi t$)
\[
\begin{aligned}
X_{\xi t} - X(0) &= \frac{V_0-V_{\xi t}}{\xi} - \frac{1}{\xi} \int_0^{\xi t} \nabla_x \phi(X_s) \, ds + \sqrt{\frac{2}{\xi}} \, dW_ {\xi s} \\
&= \frac{V_0 - V_{\xi \, t}}{\xi} - \int_0^t \nabla_x \phi (X_{\xi \,s}) ds + \sqrt{\frac{2}{\xi}} \, dW_ {\xi s},
\end{aligned}
\]
where~$(\xi^{-1/2} W_{\xi t})_{t \geq 0}$ still is a standard $d$-dimensional Brownian motion. This suggests to introduce the limiting process~$(\overline{X}_t)_{t \geq 0}$ obtained from~$(X_{\xi t})_{t \geq 0}$ as~$\xi \to +\infty$, and highlights the fact that the relaxation time is of order~$\xi$ in this regime. 

It is in fact more convenient to perform a change of unknown function and write~$f = h \Theta$, in which case we expect~$h$ to converge in the longtime limit to the constant function with values~1. The function~$h$ satisfies the following kinetic Ornstein--Uhlenbeck equation: 
\begin{equation}\label{VOU}\tag{VOU}
  \left\{ \begin{aligned}
    & \partial_t h + \nabla_v \psi \cdot \nabla_x h - \nabla_x \phi \cdot \nabla_v h = - \, \xi \,  \nabla_v^\star \, \nabla_v h, \\
    & h(t=0,\cdot,\cdot) = h_0 \in \mathrm{L}^2(\Theta),
  \end{aligned} \right.
\end{equation}
where~$A^\star$ denotes the adjoint of a closed operator on~$\mathrm{L}^2(\Theta)$.
Equation \eqref{VOU} is linear and mass-preserving, hence we can assume the initial data to be of average zero with respect to $\Theta$ and study the long-time behaviour of the corresponding solutions to \eqref{VOU}. 
We expect that
$$\lim_{t \to \infty} \|h(t,\cdot,\cdot)\|^2_{\mathrm{L}^2(\Theta)} = 0.$$
The goal of this work is to make this limit quantitative by obtaining effective decay rates.

In the recent literature, the problem of making the last limit quantitative has been widely investigated.
Equations such as \eqref{VOU} fall inside the abstract theory of \cite{hormander1967hypoelliptic}. From this founding paper, a recent theory originated, specialising on decay rates rather than regularity issues. This approach goes under the name of hypocoercivity. We briefly review the related literature, see \cite{bernard2020hypocoercivity} for a much more detailed account. Quantitative decay rates for some degenerate (but still hypoelliptic) diffusion equations appeared in~\cite{villani2006hypocoercive,villani2009hypocoercivity,MR2034753}. The strategy there relied on twisting the $\mathrm{H}^1$-norm in order to get an equivalent Lyapunov functional showing exponential decay in time along the flow of the equations under investigation. Then, exponential decay (at the same rate) could be recovered for the $\mathrm{H}^1$-norm, with a pre-factor $C>1$ in front arising from the twist between the reference norm and the Lyapunov functional.  

Later, a $\mathrm{L}^2-$setting for hypocoercivity was established, \cite{MR2215889,dolbeault2015hypocoercivity}. The strategy of \cite{dolbeault2015hypocoercivity} was to twist the standard $\mathrm{L}^2$-norm with a well-suited term in order to find an equivalent norm with a better dissipation estimate along kinetic PDEs. In addition, in this setting, one can study consistency with hydrodynamic limits and non-regularising kinetic PDEs with general initial data. The $\mathrm{L}^2$-framework is general enough to treat a large class of models~\cite{bouin2017hypocoercivity,bouin2019fractional,bouin2019hypocoercivity}, and it can provide sharp estimates for some specific models \cite{arnold2014sharp,achleitner2016linear,achleitner2017optimal,achleitner2017multi,arnold2021}. 
Precise algebraic features of the framework and estimates with respect to the parameters defining the dynamics were studied in \cite{bernard2020hypocoercivity}. In particular, the scaling with respect to $\xi$ has been also treated in \cite{GROTHAUS20143515,gotz2007stochastic}.

In order to complete this short review, let us mention also a few other works related to hypocoercive techniques, namely \cite{eberle2019couplings, dietert2015convergence} for an approach based on couplings, \cite{baudoin2013bakryp} for a version of the Bakry--\'Emery methods for hypocoercive dynamics, and \cite{dietert2022quantitative} which recently studied a model with further degeneracy.   

In the present paper, we focus on the approach of \cite{albritton2019variational}, whose techniques work directly in the standard $\mathrm{L}^2$-norm. This is possible thanks to an adapted space-time-velocity Poincar\'e inequality. We notice that time-integrated functionals of the solutions to kinetic equations are in use since \cite{guo2002landau, strain2004stability} at least. 
The framework of \cite{albritton2019variational} was qualitative, although, constructive estimates can be derived in special cases \cite{brigati2022time,cao2019explicit}. 
One idea common to both references is to exploit a version of a space--time Poincar\'e-Lions inequality~\cite{amrouche2015lemma,carrapatoso2022weighted,dietert2022quantitative}, coupled with an explicit version of the computations in \cite[Proposition 6.2]{albritton2019variational}. 

In this paper we further generalise the techniques of \cite{brigati2022time,cao2019explicit}, still keeping all estimates fully constructive. 
The main original contributions are:
\begin{enumerate}
    \item Adapting the Armstrong--Mourrat technique, we get fully explicit convergence estimates for \eqref{VOU} with general potentials $\phi,\psi$. In \cite{brigati2022time} the spatial variable was confined in a torus with $\phi=0,$ while \cite{cao2019explicit} only considers quadratic kinetic energies. Decay rates are shown to be exponential if $\psi$ grows fast enough at infinity. Otherwise, algebraic decay rates are obtained. 
    \item The potentials $\phi$ we consider are rather general. In particular, we remove the condition that the embedding $\mathrm{H}^1(\mu) \hookrightarrow \mathrm{L}^2(\mu)$ is compact, see \cite[Assumption 3]{cao2019explicit}.
    \item Our estimates for Theorem \ref{thm2} should be compared with those of \cite{dietert2022quantitative}. While performed in a simpler geometry, our result is fully constructive, as we actually build a solution for the elliptic equation involved to obtain the result . 
    \item We establish a theory for Lyapunov functionals for \eqref{VOU}, depending on the behaviour of $\psi$ at infinity. 
    \item Decay estimates are fully explicit in the fluctuation--collision parameter $\xi$.
\end{enumerate}
Some key-points of our method are the following.
\begin{itemize}
    \item We use time averages of the $\mathrm{L}^2$-norm of the solutions to \eqref{VOU} as entropies. Time averages allow for an effective entropy-entropy production estimate along the flow of \eqref{VOU}, where the key step in the proof is an adapted time-space-velocity Poincar\'e inequality.
    \item If $\psi$ grows at least linearly at infinity, the entropy production is strong enough to allow the use of a Gronwall lemma. Otherwise, the dissipation controls just a power of the time average of the norm. Then, an algebraic decay is obtained via a Bihari--Lasalle argument, combined with a uniform bound in time for a momentum of the solution. 
    \item A fundamental ingredient is a space-time weighted Lions' inequality, whose proof is fully constructive. Indeed, the problem is reduced to a regularity estimate for some elliptic PDE, which is solved either by explicit computations or with the Lax--Milgram theorem, depending on the nature of the source term in the elliptic PDE.  
\end{itemize}

\subsubsection*{Extensions.} As mentioned earlier, our aim in this work is to present a general methodology to obtain decay rates for Fokker--Planck type equations which are fully explicit in the various parameters of the dynamics, in particular the friction and the dimension. Our presentation is as general as possible, so that extensions and adaptations to various other dynamics can be worked out. We have in mind here the linear Boltzmann equation, the linearised Landau equation~\cite{Rachid20}, Fokker--Planck equations with fractional Laplacians, generalized Langevin dynamics~\cite{OP11}, Adaptive Langevin dynamics~\cite{LSS19}, etc. In fact, as our approach makes only a limited use of the linearity of the Fokker--Planck equation at hand, generalizations to nonlinear equations can be envisioned.

\subsubsection*{Outline of the paper.}
This work is organized as follows. We start in Section~\ref{sec:main_results} by presenting the problem we consider and stating the main results we establish. We give in Section~\ref{sec:preliminary} some preliminary results which are useful for the proofs of the main results. Section~\ref{sec:sec3} is devoted to the proof of exponential decay estimates where the kinetic energy $\psi$ grows linearly or faster at infinity. Section~\ref{sec:sec4} proves algebraic decay rates for solutions to \eqref{VOU} driven by kinetic energies $\psi$ which grow slower than linearly at infinity. 
Finally, weighted Lions estimates are proved in  Section \ref{sec:sec5}. Complementary results are collected in the Appendix. 


\section{Main results}
\label{sec:main_results}

This section contains the main assumptions (Section~\ref{mr:ss1}) and results (Ssections~\ref{mr:ss15}, \ref{mr:ss2}, \ref{mr:ss3}, and~\ref{mr:ss4}) of the paper. 

\subsection{Structural assumptions}\label{mr:ss1}
In order to present our main convergence result, we introduce a time~$\tau > 0$, and the associated uniform probability measure on the time interval~$[0,\tau]$:
\[
\mathrm{U}_\tau(dt) = \frac{1}{\tau} \mathbf{1}_{[0,\tau]}(t) \, dt.
\]
With some abuse of notation we sometimes denote probability measures and their densities with the same symbol.
We state the main assumptions we need for the potentials~$\phi,\psi$. 
\begin{assumption}\label{ass1}
The function $\phi$ is smooth and $\mathrm{e}^{-\phi} \in \mathrm{L}^1(dx),$ with $\int_{\mathscr{X}} \mathrm{e}^{-\phi} \, dx =1$. Moreover, for any time $\tau>0$, there exists a constant  $C^\mathrm{Lions}_\tau>0$ such that the following Lions inequality holds true:
 \begin{equation}
    \label{lionstx}
    \forall g \in \mathrm{L}^2(\mathrm{U}_\tau \otimes \mu), \quad \left\|g- \iint_{[0,\tau] \times \mathscr{X}} g(t,x) \, \mathrm{U}_\tau(dt) \otimes \mu(dx)\right\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)} \leq C_\tau^\mathrm{Lions} \|\nabla_{t,x} g  \|^2_{\mathrm{H}^{-1}(\mathrm{U}_\tau \otimes \mu)}.
\end{equation}
\end{assumption}

Sufficient conditions for \eqref{lionstx} to  hold with explicit estimates of $C^{\mathrm{Lions}}_\tau$ are discussed in Section \ref{mr:ss4}.

\begin{assumption}\label{ass2}
The function $\psi$ is smooth and $\mathrm{e}^{-\psi} \in \mathrm{L}^1(dv)$ with $\int_{\mathscr V} \mathrm{e}^{-\psi} \, dv =1$. Moreover,
\begin{equation}
    \label{eq:integrability_conditions_on_psi}
    \int_{\mathcal{V}} |\nabla_v \psi(v)|^4 \, \gamma(dv) +
    \int_{\mathcal{V}} \left|\nabla_v^2 \psi(v)\right|^2 \, \gamma(dv) < +\infty, \quad \limsup_{R \to \infty} \, \int_{\mathscr V \setminus B(0,R)} |\nabla \psi|^{-2} \, \gamma(dv) = 0,
  \end{equation}
  and the symmetric positive matrix
  \begin{equation}
    \label{eq:matrix_M}
    \mathscr{M} = \int_{\mathcal{V}} \nabla_v \psi \otimes \nabla_v \psi \, d\gamma
  \end{equation}
is definite.
\end{assumption}

The condition~\eqref{eq:matrix_M} is satisfied when~$\mathcal{V} = \mathbb{R}^d$, see~\cite[Appendix~A]{ST18} for a proof. More generally, this condition holds as soon as there is no direction~$\rho \in \mathbb{R}^d$ such that~$\rho^\top \nabla\psi(v) = 0$ for all~$v \in \mathcal{V}$. The integrability conditions~\eqref{eq:integrability_conditions_on_psi} are rather easy to satisfy. The estimates hold for instance if~$\mathcal{V}$ is bounded, or~$\psi$ increases polynomially or logarithmically at infinity. In the last case $\psi$ shall grow fast enough so that $\gamma$ is integrable and has a finite momentum of order two, see Section \ref{subsec:prop1new}.  

\begin{remark}
  The analysis we perform can straightforwardly be extended, at least from an algebraic viewpoint, to other types of degenerate symmetric dissipation operators, such as the one appearing in the linear Boltzmann equation, or involving fractional Laplacians, see \cite{dolbeault2015hypocoercivity,bouin2019fractional}.
\end{remark}

\begin{assumption}\label{ass3}
The Lie algebra generated by $\nabla_v$ and $\nabla_v \psi \cdot \nabla_x  - \nabla_x \phi \cdot \nabla_v$ is of dimension~$2d$ at all points $(x,v) \in \mathcal{X} \times \mathcal{V}$.
\end{assumption}


\subsection{An averaging lemma}\label{mr:ss15}
Averaging lemmata are a typical tool in kinetic theory to recover increased regularity on the velocity average of the distribution function. 
This is the idea behind our first result, which will be fundamental for the main theorems in Sections~\ref{mr:ss2} and \ref{mr:ss3}. 

We make a further assumption on $\phi,$ which is satisfied (with explicit control on the constant) in many cases of interest, see ~\cite[Lemma~A.24]{villani2009hypocoercivity},  as well as~\cite[Lemma~2.2]{cao2019explicit} and~\cite[Lemma~3.7]{bernard2020hypocoercivity} for a precise quantification of the dependence of the constant $L_\phi$ on the dimension, under certain structural assumptions on the potential $\phi$. This is recalled in Lemma~\ref{lemmbfls} below.

\begin{assumption}\label{ass1bis}
There exists a constant $L_\phi \in \mathbb R_+$ such that
\begin{equation}\label{commphi}
    \|z \, \nabla \phi \|_{\mathrm{L}^2(\mathrm{U}_\tau  \otimes \mu)} \leq L_\phi \|z\|_{\mathrm{H}^1(\mathrm{U}_\tau \otimes \mu)}, \qquad \forall z \in \mathrm{H}^1(\mathrm{U}_\tau \otimes \mu).
\end{equation}
\end{assumption}
A functional space which will be useful in the proofs is the kinetic space, defined as follows for a positive time $\tau>0$:
\[
\mathrm{H}^1_{\mathrm{kin}} = \left\{ h \in \mathrm{L}^2\left(\mathrm{U}_\tau \otimes \mu, \mathrm{H}^1(\gamma)\right) \ \middle| \ (\partial_t + T)h \in \mathrm{L}^2\left(\mathrm{U}_\tau \otimes \mu, \mathrm{H}^{-1}(\gamma)\right) \right\}.
\]
We introduce the following projector on~$\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)$:
\[
\Pi h = \int_{\mathcal{V}} h(\cdot,\cdot,v)  \,\gamma(dv).
\]
We denote by $\mathrm{L}^2_0(\mathrm{U}_\tau \otimes \Theta)$ the space of functions in $\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)$ with zero average. Similar zero-average spaces are defined in the same way.
We are now in position to state the following useful averaging lemma for \eqref{VOU}.
\begin{lemma}
  \label{avglem}
  Under Assumptions~\ref{ass2} and~\ref{ass1bis}, there exists an explicit constant $K_\mathrm{avg} \in \mathbb{R}_+$ such that, for any~$h \in \mathrm{H}^1_{\mathrm{kin}} \cap \mathrm{L}^2_0(\mathrm{U}_\tau \otimes \Theta)$,
  \[
  \|\nabla_{t,x} \Pi\, h\|^2_{\mathrm{H}^{-1}(\mathrm{U}_\tau \otimes \mu)} \leq K_\mathrm{avg} \left( \|(\mathrm{Id}-\Pi)h\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)} + \|(\partial_t+T)h \|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu, \mathrm{H}^{-1}(\gamma))} \right).
  \]
\end{lemma}
An explicit upper bound on $K_{\mathrm{avg}}$ can be obtained, see \eqref{kavg}.   


\subsection{Exponential convergence rates} \label{mr:ss2}

We consider the longtime behaviour of solutions to \eqref{VOU}. In particular, our main goal is to prove that 
$$h(t,\cdot,\cdot) \xrightarrow[t \to \infty]{} \iint_{\mathscr{X} \times \mathscr{V}} h_0(x,v) \, \Theta(dx\,dv),$$
for all solutions to \eqref{VOU}, with quantitative and explicit convergence rates. 
As explained in Section \ref{intro}, this reduces to studying the decay to $0$ for solutions to~\eqref{VOU} with initial condition
\[
h_0 \in \mathrm{L}^2_0(\Theta) = \left\{ g \in \mathrm{L}^2(\Theta) \ \middle| \ \int_{\mathcal{X} \times \mathcal{V}} g \, d\Theta = 0\right\}.
\]
Other sub-spaces such as $\mathrm{L}^2_0(\gamma)$ are defined in a similar way.
In order to achieve exponential decay rates, we need the following. 
\begin{assumption}\label{ass4}
The following Poincar\'e inequality for~$\gamma$ holds: there exists $c_\psi>0$ such that 
\begin{equation}
  \label{eq:coercivity_Delta_psi_OU}
  \forall g \in \mathrm{H}^1(\gamma) \cap \mathrm{L}^2_0(\gamma), \qquad \|g\|_{\mathrm{L}^2(\gamma)} \leq c_\psi^{-1/2} \|\nabla_v g \|_{\mathrm{L}^2(\gamma)}.
\end{equation}
\end{assumption}
A sufficient condition on~$\psi$ for the latter inequality to hold is that there exists a constant $a \in (0,1)$ such that
$$ \liminf_{|v| \to \infty} \left( a|\nabla_v \psi(v)|^2 - \Delta_v \psi(v) \right) > 0,$$
see~\cite{persson1960bounds,10.1214/ECP.v13-1352}. The inequality~\eqref{eq:coercivity_Delta_psi} is always true when~$\mathcal{V} = \mathbb{T}^d$ and~$\psi$ is smooth. 

As in \cite{brigati2022time}, following up on ideas used in \cite{guo2002landau,albritton2019variational} we introduce, for any $h \in \mathrm{L}^2([0,\infty); \mathrm{L}^2(\Theta)),$ the following functional:
\begin{equation}\label{timeavg}
\mathscr{H}_\tau(t) = \int_t^{t+\tau} \|h(s,\cdot,\cdot)\|^2_{\mathrm{L}^2(\Theta)} ds.
\end{equation}
Then, our main result goes as follows. 
\begin{theorem}\label{thm1new}
Under Assumptions \ref{ass1}-\ref{ass2}-\ref{ass3}-\ref{ass1bis}-\ref{ass4}, there exists a constant $\lambda>0$ such that all functions $h$ solving \eqref{VOU} satisfy  
\[
\forall \, t \geq 0, \qquad \mathscr{H}_\tau(t) \leq  \mathrm{e}^{-2\lambda \, t} \mathscr{H}_\tau(0).
\]
Moreover, 
\[
\lambda \geq  \bar{\lambda} \, \left( \frac{1}{\xi c_\psi} + \xi \right)^{-1}, \qquad \bar{\lambda} = \left( 1 + C^{\mathrm{Lions}}_\tau \, K_\mathrm{avg} \right)^{-1}.
  \]
\end{theorem}

Theorem \ref{thm1new} implies a standard hypocoercivity estimate (see \cite{villani2009hypocoercivity}) for solutions to \eqref{VOU}. Note that there is no prefactor $C$ in the exponential decay estimate of Theorem \ref{thm1new}. Pointwise in time decay estimates can be deduced from Theorem \ref{thm1new} upon adding such a prefactor.

\begin{corollary}\label{cor1new}
Under the same assumptions as Theorem \ref{thm1new}, all solutions to \eqref{VOU} satisfy
$$\forall t \geq 0, \qquad  \|h(t,\cdot,\cdot)\|^2_{\mathrm{L}^2(\Theta)} \leq C \, \mathrm{e}^{-\lambda t} \, \|h_0\|^2_{\mathrm{L}^2(\Theta)},$$
with~$C = \mathrm{e}^{\lambda \tau}$.
\end{corollary}

\subsection{Algebraic convergence rates}\label{mr:ss3}

Condition \eqref{eq:coercivity_Delta_psi_OU} is equivalent to the fact that the symmetric part $\mathcal{S}$ is coercive, in the following sense:
\begin{equation}
\label{eq:coercivity_Delta_psi}
  \mathcal{S} \geq c_\psi \mathbb{1} \quad \text {on} \quad  \mathrm{L}^2_0(\gamma).
\end{equation}
Typical cases where the last condition (i.e. Assumption \ref{ass4}) does not hold correspond to a kinetic energy $\psi$ growing less than linearly at infinity, such as
\begin{equation}\label{exmp1}
    \psi = (1+|v|^2)^{\frac{\alpha}{2}}, \quad \alpha \in (0,1), 
\end{equation}    
or    
\begin{equation}\label{exmp2}    \psi(v) = \frac{\beta}{2} \, \log \left(1+|v|^2\right) , \quad \beta >d+2, 
\end{equation}  
so that $\gamma$ is heavy tailed at infinity. However, these classes of examples satisfy the following weighted Poincar\'e inequality (see \cite{blanchet2007hardy,bouin2019hypocoercivity}) that we state in a general form in Assumption \ref{ass5} below.

\begin{assumption}\label{ass5}
There exist a constant $P_{\psi} \in \mathbb R_+$, and a function $\mathscr G \, \in \mathrm{L}^1(\gamma)$ such that 
\begin{equation}\label{eq:weightedpoi}
    \forall g \in \mathrm{H}^1(\gamma) \cap \mathrm{L}^2_0(\gamma), \qquad \int_{\mathbb R^d} \mathscr G^{-1}(v) \, g^2(v) \, \gamma(dv) \leq P_\psi \|\nabla_v g \|^2_{\mathrm{L}^2(\gamma)}.
\end{equation}
\end{assumption}

For the classes of examples \eqref{exmp1}-\eqref{exmp2}, we can respectively consider $\mathscr G(v) = (1+|v|^2)^{1-\alpha}$ and $\mathscr G(v) = 1+|v|^2$, see~\cite{blanchet2007hardy,bouin2019hypocoercivity}.

In the cases covered by Assumption \ref{ass5}, we recover constructive algebraic decay rates for~$\mathscr H_\tau$ as follows.

\begin{theorem}\label{thm2new}
Suppose that Assumptions \ref{ass1}-\ref{ass2}-\ref{ass3}-\ref{ass1bis}-\ref{ass5} hold, and consider $\sigma > 0$ such that $\mathscr G^\sigma \in \mathrm{L}^1(\gamma)$. If all solutions $h$ to \eqref{VOU} satisfy
\begin{equation}\label{moments}
\iint_{\mathscr{X} \times \mathscr{V}} \mathscr G(v)^\sigma \, h^2(t,x,v) \, \Theta(dx \, dv) \leq A \mathscr U (h_0), \qquad \forall t \geq 0,
\end{equation}
for a functional $\mathscr{U} : \mathrm{L}^2(\Theta) \to [0,\infty]$ and a constant $A(\psi,\sigma,\phi)>0$, then there exist two explicit constants $c_1(h_0),c_2(h_0) \in \mathbb R_+$ depending only on  $\|h_0\|^2_{\mathrm{L}^2(\Theta)}, \mathscr{U}(h_0), \phi, \psi, \sigma,$ and $d$ such that
    \[
    \forall \xi > 0, \quad \forall t \geq 0, \qquad   \mathscr{H}_\tau(t) \leq \frac{\mathscr{H}_\tau(0)}{\left( 1 + \left( \xi^{-\frac{\sigma}{\sigma+1}} c_1 + \xi c_2 \right)^{-\frac{\sigma+1}{\sigma}} t \right)^{\sigma}},
    \]
\end{theorem}
Explicit expressions for $c_1,c_2$ can be obtained, see Section \ref{sec:proof_thm1}. 
As in Section \ref{mr:ss3}, a decay estimate for $\mathscr H$ implies a point-wise decay estimate in time. We state a result analogous to the one of Corollary \ref{cor1new} in the present situation.

\begin{corollary}\label{cor2new}
Under the same assumptions as Theorem \ref{thm2new}, all solutions to \eqref{VOU} satisfy
$$\forall t \geq 0, \qquad \|h(t,\cdot,\cdot)\|^2_{\mathrm{L}^2(\Theta)} \leq \frac{\|h_0\|^2_{\mathrm{L}^2(\Theta)}}{\left( 1 + \left( \xi^{-\frac{\sigma}{\sigma+1}} c_1 + \xi c_2 \right)^{-\frac{\sigma+1}{\sigma}} (t+\tau) \right)^{\sigma}}.$$
\end{corollary}

\begin{remark}
Note that Theorems \ref{thm1new} and \ref{thm2new} are qualitatively consistent. Indeed, the first one can be recovered from the second one in the regime corresponding to a sequence of sub-exponential local equilibria $\gamma$ becoming exponential in the limit. In this case, \eqref{eq:weightedpoi} holds with $\mathscr G \to 1$, so that the choice $\sigma \to \infty$ is allowed, yielding a decay which is faster than any inverse power of time, hence no longer algebraic.
\end{remark}

\begin{remark}
An alternative approach to obtaining a convergence result similar to Theorem \ref{thm2new} is that of weak Poincar\'e inequalities, see \cite{grothaus2019weak}, when the initial data of \eqref{VOU} are $\mathrm{L}^\infty$ functions. In this case, the analysis on Lyapunov functions we perform in Section \ref{sec:proof_thm1} would be substituted by the maximum principle for hypoelliptic operators, see \cite{taira2019strong}. \end{remark}

Theorem \ref{thm2new} crucially relies on the weighted $\mathrm{L}^2$ bound \eqref{moments}, which we now motivate and discuss. 
When the local equilibrium $\gamma$ decays slower than exponentially at infinity, the  phenomenon of \textit{loss of velocity moments} occurs.
This means that the $\mathrm{L}^2-$norm of the solution cannot be controlled without assuming the control of extra velocity moments for the initial data. In addition, we have to ensure that these moments are propagated by \eqref{VOU}. Such an issue is present even in the spatially-homogeneous case. 
This has already been studied in \cite{bouin2019hypocoercivity,brigati2022time,carrapatoso2017landau}. 
In order to follow the strategy of proof of \cite{bouin2019hypocoercivity,brigati2022time}, the term we need to bound along the flow of \eqref{VOU} reads 
\begin{equation}\label{momentumg}
\iint_{\mathscr{X} \times \mathscr{V}} \mathscr G(v)^\sigma \, h^2(t,x,v) \, \Theta(dx \, dv).    
\end{equation}
However, obtaining a direct uniform control is tricky in practice. Hence, our strategy is to control \eqref{momentumg} with a Lyapunov functional $\mathscr{U}$, whose evolution along the flow of \eqref{VOU} is easier to determine, thanks to a control at infinity, as stated precisely in \eqref{eq:lyapunovabstr}.

In fact, an explicit expression for $\mathscr U$ can be found under some growth assumptions on $\psi$, as made precise in the following result. 

\begin{proposition}\label{prop1new}
Assume that $\psi(v) \to \infty,$ when $|v| \to \infty,$ and $\phi(x) \to \infty$ when $|x| \to \infty.$ Let $\mathscr G \in \mathrm{L}^1(\gamma)$, such that Assumption \ref{ass5} holds true and fix an exponent $\sigma>0$.
\begin{enumerate}
    \item Suppose that there exist  $\eta,\ell > 0$ such that  
    \begin{equation}\label{subexponential1}
    \limsup_{|v| \to \infty} \frac{\mathscr G(v)^\sigma}{1+\psi(v)^\eta} < \infty, 
    \end{equation}
    and
    \begin{equation}\label{subexponential2} 
    \liminf_{|v| \to \infty} \, \psi(v)^{\ell-1} \left[ \left(1- \frac{\eta-1}{\psi(v)} \right) |\nabla_v \psi(v)|^2 - \Delta_v \psi(v)  \right] >0.
    \end{equation}
    Then, the functional 
    $$\mathscr{U}(g) = \iint g(x,v)^2 \, (\phi(x) + \psi(v))^\eta \, \Theta(dx \, dv),$$ satisfies \eqref{moments}.
    \item Suppose that there exist $0<\ell<\eta<1$ such that 
    \begin{equation}
        \label{fattail2}
      \limsup_{|v| \to \infty} \frac{\mathscr G(v)^\sigma}{   e^{\eta \psi(v)} + 1} < \infty, 
    \end{equation}
    and  \begin{equation}\label{fattail1}
    \liminf_{|v| \to \infty} \, \mathrm{e}^{\ell \psi(v)} \, \left( (1-\eta) |\nabla_v \psi(v)|^2 - \Delta_v \psi(v) \right) >0.
    \end{equation}
    Then, \eqref{moments} holds with $$\mathscr{U}(g) = \iint g(x,v)^2 \, \mathrm{e}^{\eta(\phi(x) + \psi(v))} \, \Theta(dx \, dv).$$ 
\end{enumerate}
\end{proposition}

\begin{remark}\label{oss33}
The classes of examples \eqref{exmp1}-\eqref{exmp2} are covered by points (1) and (2) of Proposition \ref{prop1new}, respectively, if we take the appropriate expression for $\mathscr G$. 
We give explicit lower bounds on $\eta$, see Section \ref{subsec:lyapunov}, and show that the value of $\ell$ depends only on the form of $\psi$ and not on $\eta$. 
\end{remark}

 



\subsection{Weighted Lions' inequalities}\label{mr:ss4}

Inequality \eqref{lionstx} is a generalised version of Lions' inequality~\cite{amrouche2015lemma}. Such an inequality with a weighted measure has been first studied in~\cite{carrapatoso2022weighted}. We give here a simple sufficient condition on the potential $\phi$ for~\eqref{lionstx} to hold. We also obtain an explicit estimate of $C^{\mathrm{Lions}}_\tau$, under some growth conditions on the spatial potential~$\phi$.

\begin{theorem}\label{thm2}
Assume that $\phi$ is a smooth potential such that $\int_\mathscr{X} \mathrm{e}^{-\phi} \,dx = 1$, that its associated probability measure $\mu$ satisfies the Poincar\'e inequality:
\begin{equation}\label{poiphi}
    \|\nabla_x z\|_{\mathrm{L}^2(\mu)} \geq c_\phi^{-\frac{1}{2}} \, \|z\|_{\mathrm{L}^2(\mu)}, \qquad \forall z \in \mathrm{H}^1(\mu) \cap \mathrm{L}_0^2(\mu),
\end{equation}
for some $c_\phi \in \mathbb R_+$, and that
\begin{equation}\label{hess}
|\nabla^2 \phi|^2 \leq (c'_\phi)^2 \, \left( \, d + |\nabla_x \phi|^2 \right), \qquad \Delta \phi \leq c_\phi'' \left( d +  |\nabla_x \phi|^2 \right).
\end{equation}
for some constants $c'_\phi, c''_\phi \in {\mathbb R}_+$.
Then, Assumption \ref{ass1} is satisfied, with 
$$C_\tau^{\mathrm{Lions}} = C_\phi \, \left( 1 + \sqrt{d} \right) \, \left( \tau^2 + \tau^{-2} \right)^2,$$
for a constructive constant $C_\phi$, depending only on $c_\phi,c'_\phi,c''_\phi$, but not directly on $d$.
\end{theorem}

The explicit dependence on $d$ in the estimate of $C^{\mathrm{Lions}}_\tau$, which is exactly of order~$\sqrt{d}$ if the Poincar\'e constant $c_\phi$ does not depend on $d$, is important for the case of a Boltzmann-Gibbs measure obtained as the tensorisation of several $1$-dimensional ones (as in the context of sampling methods, see~\cite{ST18,LFR19}). Dimension independent Poincar\'e constants can also be obtained for systems with \textit{finite number of interactions}, see \cite{bernard2020hypocoercivity}.
The scaling in the second inequality of \eqref{hess} is natural, as it is the correct one for separable potentials $\phi = \sum_{i=1}^d \bar{\phi}(x_i),$ with $\bar{\phi} : {\mathbb R} \to {\mathbb R}$ such that $\bar{\phi}'' \leq c''_\phi \left( 1 + |\bar{\phi}'|^2\right).$ 
Sufficient conditions for \eqref{poiphi} to hold true are given in \cite{persson1960bounds,10.1214/ECP.v13-1352}. A noteworthy case in the theorem above is when the injection $\mathrm{H}^1(\mu) \hookrightarrow \mathrm{L}^2(\mu)$ is compact. In this situation, the estimates can be inferred from \cite[Lemma 2.6]{cao2019explicit}; in particular $C_\tau^{\mathrm{Lions}}$ is bounded by the constant in \cite[Equation (27)]{cao2019explicit}.




















\section{Notation and preliminary technical results}
\label{sec:preliminary}
We denote by $A^\star$ for the adjoint of a closed operator $A$ on the Hilbert space $\mathrm{L}^2(\Theta)$, which, for any smooth functions~$h_1,h_2$ with compact supports, satisfies
\[
\int_{\mathcal{X} \times \mathcal{V}} (Ah_1) h_2 \, d\Theta = \int_{\mathcal{X} \times \mathcal{V}} h_1 \left(A^\star h_2\right) d\Theta.
\]
The action of~$A^\star$ is obtained by integration by parts for differential operators. In particular, for $1 \leq i \leq d$,
\begin{equation}
  \label{VOUo}
  \partial_{x_i}^\star = -\partial_{x_i} + \partial_{x_i} \phi,
  \qquad
  \partial_{v_i}^\star = -\partial_{v_i} + \partial_{v_i} \psi.
\end{equation}
This allows to rewrite the operators appearing in~\eqref{VOU} as the sum of a skew-symmetric operator
\[
T =: \nabla_v \psi \cdot \nabla_x - \, \nabla_x \phi \cdot \nabla_v = \sum_{i=1}^d \partial_{v_i}^\star\partial_{x_i} - \partial_{x_i}^\star\partial_{v_i}, 
\]
and a symmetric one
\begin{equation}
  \label{eq:S_decomposition}
  \mathcal{S} = -\nabla_v^\star \nabla_v = -\sum_{i=1}^d \partial_{v_i}^\star\partial_{v_i},
\end{equation}
so that~\eqref{VOU} rewrites
\[
\partial_t h + Th = \xi \mathcal{S} h.
\]
The right hand side of \eqref{regest} is indeed always finite. Solutions to \eqref{VOU} live in the Bochner space~$\mathrm{L}^2(0,\infty;\mathrm{L}^2(\mu, \mathrm{H}^1(\gamma)))$, as we show thanks to the following inequality for solutions to~\eqref{VOU}:
\begin{equation}
  \label{energy}
  \frac{d}{dt} \|h(t)\|^2_{\mathrm{L}^2(\Theta)} = 2\left\langle h(t), Th(t)\right\rangle_{\mathrm{L}^2(\Theta)} + 2\xi \left\langle h(t), \mathcal{S}h(t)\right\rangle_{\mathrm{L}^2(\Theta)}= - 2\xi \| \, \nabla_v h\|^2_{\mathrm{L}^2(\Theta)} \leq 0,
\end{equation}
in view of the skew-symmetry of~$T$ and~\eqref{eq:S_decomposition}.


We also denote by $\mathrm{H}^{-1}(\mathrm{U}_\tau \otimes \mu)$ the dual of the space
\begin{equation}\label{dcdef}
\mathrm{H}^1_\mathrm{DC}(\mathrm{U}_\tau \otimes \mu) = \left\{ h \in \mathrm{H}^{1}(\mathrm{U}_\tau \otimes \mu) \, \middle| \, h(0,\cdot) = h(\tau,\cdot) =0 \right\},
\end{equation}
where the subscript $\mathrm{DC}$ stands for \textit{Dirichlet conditions}.
Note that the boundary conditions in time are well defined because the trace of functions~$h \in \mathrm{H}^{1}(\mathrm{U}_\tau \otimes \mu)$ makes sense on the boundary~$\{0,\tau\} \times \mathcal{E}$ of the domain. 
Under Assumption \ref{ass3}, the operator~$-T + \xi \mathcal{S}$ is hypoelliptic (see~\cite{hormander1967hypoelliptic,villani2009hypocoercivity}), hence the solutions to~\eqref{VOU} are smooth. 

The first result provides estimates on the skew-symmetric part of the space-time Fokker--Planck operator~$\partial_t + T$ in terms of its symmetric part~$\mathcal{S}$, for solutions of~\eqref{VOU}.

\begin{lemma}
  Consider a solution of~\eqref{VOU}. Then, 
  \begin{equation}\label{regest}
    \|(\partial_t + T)h \|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu ; \mathrm{H}^{-1}(\gamma))} \leq \xi \| \nabla_v h\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)}.
  \end{equation}
  In particular, solutions to~\eqref{VOU} belong to $\mathrm{H}^1_{\mathrm{kin}}.$
\end{lemma}

\begin{proof}
  Fix a test function $u \in \mathrm{L}^2(\mathrm{U}_\tau \otimes \mu ; \mathrm{H}^1(\gamma))$ such that $\|u\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu ; \mathrm{H}^1(\gamma))} \leq 1$.
  Then, denoting by~$\langle \, \cdot,\cdot \,\rangle_{\mathrm{L}^2(\mathrm{H}^{-1}),\mathrm{L}^2(\mathrm{H}^1)}$ the duality bracket between $\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu ; \mathrm{H}^{-1}(\gamma))$ and $\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu ; \mathrm{H}^1(\gamma))$, 
\[
\left\langle (\partial_t  + T)h , u \right\rangle_{\mathrm{L}^2(\mathrm{H}^{-1}),\mathrm{L}^2(\mathrm{H}^1)} = \xi \left\langle \mathcal{S} h , u \right\rangle_{\mathrm{L}^2(\mathrm{H}^{-1}),\mathrm{L}^2(\mathrm{H}^1)} = - \int_0^\tau \int_{\mathcal{X} \times \mathcal{V}} \nabla_v h \cdot \nabla_v u \, d\mathrm{U}_\tau \, d\Theta.
\]
The desired estimate follows from first bounding the right hand side via the Cauchy-Schwarz inequality, and then taking the supremum over~$u$.
\end{proof}




The next lemma, whose proof is omitted, provides estimates similar to \cite[Lemma~2.3]{cao2019explicit}, obtained with an adaptation of the approach of \cite[Lemma 3.6 and~3.7]{bernard2020hypocoercivity} to spacetime operators. It provides a fully explicit dependence of the estimates on the dimension~$d.$

\begin{lemma}\label{lemmbfls}
Assume that $\phi$ is a smooth potential such that $\int_\mathscr{X} \mathrm{e}^{-\phi} \,dx = 1$, and   \eqref{poiphi}-\eqref{hess} hold true. Then, Assumption \ref{ass1bis} holds true with 
$$L_\phi^2 = \max \, \left(16 , 4 \, d\, c''_\phi \right).$$
Moreover, for any $z \in \mathrm{H}^2(\mathrm{U}_\tau \otimes \mu)$ such that $\nabla_{t,x} z \in \mathrm{H}^1_{\mathrm{DC}}(\mathrm{U}_\tau \otimes \mu; \mathbb R^d),$
\begin{equation}
    \label{bochnerx}
    \|\nabla_{t,x}^2 z \|_{\mathrm{L}^2\left(\mathrm{U}_\tau \otimes \mu\right)}^2 \leq 2 \|\nabla_{t,x}^\star \nabla_x z \|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)} + 2 c'_\phi \left(\sqrt{d} + 2 \max \left(8 c'_\phi , \sqrt{c''_\phi \,d} \right)  \,  \right) \,  \|\nabla_{t,x} z \|^2_{\mathrm{L}^2\left(\mathrm{U}_\tau \otimes \mu\right)}.
\end{equation}
\end{lemma}
It would be possible to have better constants in \eqref{bochnerx} under additional assumptions, for example if $\phi$ is strongly convex, see \cite{bernard2020hypocoercivity,cao2019explicit}. 

\section{Modified Poincar\'e inequalities and exponential decay rates}\label{sec:sec3}

We start by proving Lemma \ref{avglem} in Section~\ref{sec:avglem}, using a Poincar\'e-type inequality.
We turn to the proof of Theorem \ref{thm1new} in Section~\ref{sec:proof_thm1}. 
We finally prove Corollary \ref{cor1new} in Section~\ref{subseccor1}. 


\subsection{Proof of Lemma \ref{avglem} and a Poincar\'e-like inequality}\label{sec:avglem}


The important Poincar\'e-type inequality which is used to prove Theorem~\ref{thm1new} is the following. Its proof crucially relies on Lemma \ref{avglem}. 

\begin{proposition}
  \label{prop:Poincare_type}
Under Assumptions \ref{ass1}-\ref{ass2}-\ref{ass1bis}, it holds
\begin{equation}
  \label{modpoi}
  \forall h \in \mathrm{H}^1_{\mathrm{kin}} \cap \mathrm{L}^2_0(\mathrm{U}_\tau \otimes \Theta),
  \qquad 
  \bar{\lambda} \| h\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes\Theta)} \leq  \|(\partial_t + T)h\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu;\mathrm{H}^{-1}(\gamma))} + \|(\mathrm{Id}- \Pi)h\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)},
\end{equation}
with
$$\bar{\lambda} = \left( 1 + C_\tau^{\mathrm{Lions}} \, K_\mathrm{avg} \right)^{-1},$$
where $K_{\mathrm{avg}}$ is the constant appearing in Lemma \ref{avglem}.
\end{proposition}

\begin{proof}[Proof of Proposition~\ref{prop:Poincare_type}]
  Since~$\Pi$ is an orthogonal projector, 
  \begin{equation}\label{wpoiint}
  \|h\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)} = \|(\mathrm{Id}-\Pi)h\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)} + \|\Pi h\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)}.
  \end{equation}
  We apply~\eqref{lionstx} to the second term on the right hand side in \eqref{wpoiint}, and then use Lemma \ref{avglem} to write
  \[
  \begin{aligned}
  \|\Pi h\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)} & \leq  C^{\mathrm{Lions}}_\tau \, \|\nabla_{t,x} \Pi h\|^2_{\mathrm{H}^{-1}(\mathrm{U}_\tau \otimes \mu)} \\
  &\leq C^{\mathrm{Lions}}_\tau \, \mathrm{K}_{\mathrm{avg}} \left( \|(\mathrm{Id}-\Pi)h\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)} + \|(\partial_t+T)h \|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu, \mathrm{H}^{-1}(\gamma))} \right).
  \end{aligned}
  \]
  Plugging the last estimate into \eqref{wpoiint} provides the claimed result.
\end{proof}

We conclude this section with the proof of Lemma~\ref{avglem}.

\begin{proof}[Proof of Lemma~\ref{avglem}]
  Some integrals in the calculation below are formal and represent duality products. In order to estimate the~$\mathrm{H}^{-1}$ norm of~$\nabla_{t,x} \Pi h$, we first compute the $\mathrm{H}^{-1}$ norm of~$\partial_t \Pi h$, and then turn to~$\nabla_x \Pi h$. To bound~$\|\partial_t \Pi h\|_{\mathrm{H}^{-1}(\mathrm{U}_\tau \otimes \mu)}$, we fix a test function~$z \in \mathrm{H}^1_\mathrm{DC}(\mathrm{U}_\tau \otimes \mu)$ with $\|z\|^2_{\mathrm{H}^1(\mathrm{U}_\tau \otimes \mu)} \leq 1$; while the estimation of~$\|\nabla_x \Pi h\|_{\mathrm{H}^{-1}(\mathrm{U}_\tau \otimes \mu)}$ can be performed by considering a test function~$Z =(Z_1,\dots,Z_d)\in \mathrm{H}^1(\mathrm{U}_\tau \otimes \mu)^d$ with $\|Z\|^2_{\mathrm{H}^1(\mathrm{U}_\tau \otimes \mu)} \leq 1$.
  
  For~$\partial_t \Pi h$, we start by noticing that $\nabla_v \Pi h = 0$ and
  \[
  \int_{\mathcal{V}} \nabla \psi(v) \, \gamma(dv) = 0,
  \]
  so that, using the skew-symmetry of $\partial_t + T$ and the Dirichlet boundary conditions for the second term, 
  \[
  \begin{aligned}
    & \int_0^\tau \int_{\mathcal{X}} (\partial_t \Pi h) \, z \, d\mathrm{U}_\tau \, d\mu = \int_0^\tau \int_{\mathcal{X}} \int_{\mathcal{V}} ((\partial_t + T) \Pi \, h)(t,x,v) \, z(t,x) \, \mathrm{U}_\tau(dt)\, \mu(dx) \,\gamma(dv) \\
    & \qquad = \int_0^\tau \int_{\mathcal{X}} \int_{\mathcal{V}} [(\partial_t + T) h]  \, z \, d\mathrm{U}_\tau \, d\Theta + \int_0^\tau \int_{\mathcal{X}} \int_{\mathcal{V}} [(\mathrm{Id}-\Pi)h] \, (\partial_t + T) z \, d\mathrm{U}_\tau \, d\Theta \\
    & \qquad \leq \|(\partial_t + T) h\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu,\mathrm{H}^{-1}(\gamma))} \|z\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu,\mathrm{H}^1(\gamma))} + \|(\mathrm{Id}-\Pi)h\|_{\mathrm{\mathrm{L}^2}(\mathrm{U}_\tau \otimes \Theta)} \|(\partial_t + T)z\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)}.
  \end{aligned}
  \]
  The last factor in the last inequality can be bounded as 
  \[
  \|(\partial_t + T)z\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)} \leq \| \partial_t z \|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)} + \| T z\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)} \leq 1 + \|\nabla_v \psi\|_{\mathrm{L}^2(\gamma)},
  \]
  where we used $\|z\|_{\mathrm{H}^1(\mathrm{U}_\tau \otimes \mu)} \leq 1$ and a Cauchy--Schwarz inequality to write
  \[
  \| T z\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)}^2 = \| \nabla \psi \cdot \nabla_x z\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)}^2 \leq \|\nabla_x z\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)} \int_0^\tau \int_{\mathcal{X}} \int_{\mathcal{V}} |\nabla \psi(v)|^2 \, \mathrm{U}_\tau(dt) \, \mu(dx) \, \gamma(dv).
  \]
  By taking the supremum over functions~$z \in \mathrm{H}^1_\mathrm{DC}(\mathrm{U}_\tau \otimes \mu)$ with~$\|z\|^2_{\mathrm{H}^1(\mathrm{U}_\tau \otimes \mu)} \leq 1$, we finally obtain
  \[
  \|\partial_t \Pi h\|_{\mathrm{H}^{-1}(\mathrm{U}_\tau \otimes \mu)} \leq \|(\partial_t + T) h\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu,\mathrm{H}^{-1}(\gamma))} + \left(1 + \|\nabla_v \psi\|_{\mathrm{L}^2(\gamma)}\right)\|(\mathrm{Id}-\Pi)h\|_{\mathrm{\mathrm{L}^2}(\mathrm{U}_\tau \otimes \Theta)} .
  \]
  
  We next turn to the estimation of~$\|\nabla_x \Pi h\|_{\mathrm{H}^{-1}(\mathrm{U}_\tau \otimes \mu)}$. We introduce
  \[
  G = \frac{\nabla_v \psi}{\|\nabla_v \psi\|_{\mathrm{L}^2(d\gamma)}}.
  \] 
  In view of the definition~\eqref{eq:matrix_M}, it holds by construction that
  \[
  \int_\mathcal{V} G(v) \otimes M G(v) \, \gamma(dv) = \mathrm{Id}_{d\times d}, \qquad M = \|\nabla_v \psi\|_{\mathrm{L}^2(d\gamma)}^2 \mathscr{M}^{-1}. 
  \]
  The quantity to estimate is 
  \begin{equation}
    \label{eq:series_eq_nabla_x_Pi_Z}
  \begin{aligned}
   & \int_0^\tau \int_{\mathcal{X}} \nabla_x \Pi h \cdot Z \, d\mathrm{U}_\tau \, d\mu = \int_0^\tau \int_{\mathcal X} \int_{\mathcal{V}} \left[G(v) \cdot \nabla_x \Pi h(t,x)\right]\left[MG(v) \cdot Z(t,x)\right] \mathrm{U}_\tau(dt) \, \Theta(dx \, dv) \\
    & \qquad \qquad = \|\nabla_v \psi\|^{-1}_{\mathrm{L}^2(\gamma)}\int_0^\tau \int_{\mathcal X} \int_{\mathcal{V}} \left[(\partial_t + T)\Pi h\right](t,x,v)\left[MG(v) \cdot Z(t,x)\right] \mathrm{U}_\tau(dt) \, \Theta(dx \, dv) \\
    & \qquad \qquad = \|\nabla_v \psi\|^{-1}_{\mathrm{L}^2(\gamma)}\int_0^\tau \int_{\mathcal X} \int_{\mathcal{V}} \left[(\partial_t + T)h\right](t,x,v)\left[MG(v) \cdot Z(t,x)\right] \mathrm{U}_\tau(dt) \, \Theta(dx \, dv) \\
    & \qquad \qquad \ \ - \|\nabla_v \psi\|^{-1}_{\mathrm{L}^2(\gamma)}\int_0^\tau \int_{\mathcal X} \int_{\mathcal{V}} (T(1-\Pi) h)(t,x,v)\left[MG(v) \cdot Z(t,x)\right] \mathrm{U}_\tau(dt) \, \Theta(dx \, dv) \\
    & \qquad \qquad \ \ - \|\nabla_v \psi\|^{-1}_{\mathrm{L}^2(\gamma)}\int_0^\tau \int_{\mathcal X} \int_{\mathcal{V}} (\partial_t (1-\Pi) h)(t,x,v)\left[MG(v) \cdot Z(t,x)\right] \mathrm{U}_\tau(dt) \, \Theta(dx \, dv),
    \end{aligned}
  \end{equation}
  where the second equality follows from the fact that~$\nabla_v \Pi h = 0$ and $\int_{\mathcal{V}} G \, d\gamma = 0$. Since
  \[
  \|MZ\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)} \leq \|M \|_{\mathcal{B}(\ell^2)} \|Z\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)} \leq \|M \|_{\mathcal{B}(\ell^2)} \|Z\|_{\mathrm{H}^1(\mathrm{U}_\tau \otimes \mu)} \leq \|M \|_{\mathcal{B}(\ell^2)},
  \]
  where~$\|M \|_{\mathcal{B}(\ell^2)}$ denotes the matrix norm induced by the Euclidean norm, and~$\|G\|_{\mathrm{H}^1(\gamma)} < +\infty$ (in view of the second condition in~\eqref{eq:integrability_conditions_on_psi}), the first integral in the last equality can be bounded by~$\|G \cdot MZ\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu,\mathrm{H}^1(\gamma))} \|(\partial_t +T)h \|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu, \mathrm{H}^{-1}(\gamma) )} \leq \|M \|_{\mathcal{B}(\ell^2)} \|G \|_{\mathrm{H}^1(\gamma)} \|(\partial_t +T)h \|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu, \mathrm{H}^{-1}(\gamma) )}$. Since~$T$ is antisymmetric, the second integral in the last equality can be controlled as 
  \[
  \left| \int_0^\tau \int_{\mathcal X} \int_{\mathcal{V}} \left[(1-\Pi) h \right] \left[ T(MG \cdot Z)\right] d\mathrm{U}_\tau \, d\Theta \right| \leq \|(1-\Pi)h\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)} \|T(MG \cdot Z)\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)}.
  \]
  Introducing~$G^M = (G^M_1,\dots,G_d^M) = MG$ to simplify the notation,
  \[
  \begin{aligned}
    \left[ T (MG \cdot Z) \right](t,x,v)
    & = T\left[\sum_{i=1}^d G_i^M Z_i \right](t,x,v) \\
    & = \sum_{i=1}^d G_i^M(v) \nabla \psi(v) \nabla_x Z_i(t,x) - \nabla G_i^M(v) \cdot \nabla\phi(x) Z_i(t,x),
  \end{aligned}
  \]
  where~$G_i^M \nabla \psi \in \mathrm{L}^2(\gamma)$ and $\nabla G_i^M \in \mathrm{L}^2(\gamma)$ in view respectively of the first and second condition in~\eqref{eq:integrability_conditions_on_psi}; while~$Z_i \nabla \phi \in \mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)$ with~$\|Z_i \nabla \phi \|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)} \leq L_\phi \|Z_i \|_{\mathrm{H}^1(\mathrm{U}_\tau \otimes \mu)}$ by Assumption \ref{ass1bis}.
The last integral in~\eqref{eq:series_eq_nabla_x_Pi_Z} can be bounded, after transferring the partial derivative in time to the test function, by~$\|(1-\Pi)h\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)} \|MG \cdot \partial_t Z\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)}$. The second factor tensorizes as~$\|MG\|_{\mathrm{L}^2(\gamma)} \| \partial_t Z\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)} \leq \|M \|_{\mathcal{B}(\ell^2)} \|G\|_{\mathrm{L}^2(\gamma)} = \|M \|_{\mathcal{B}(\ell^2)}$. 

By gathering the above estimates, the desired result follows with the constant
\begin{equation}\label{kavg}
  \begin{aligned}
  &K_\mathrm{avg}^{\frac{1}{2}} = \\ & \max \left( 1 + \|\nabla_v \psi\|_{\mathrm{L}^2(\gamma)}  , \frac{ \|M\|_{\mathcal B(\ell^2)}\left(1+\|G\|_{\mathrm{H}^1(\gamma)}\right) + \sqrt{\sum_{i=1}^d \|G_i^M \nabla_v \psi\|^2_{\mathrm{L}^2(\gamma)}} + L_\phi \, \sqrt{\sum_{i=1}^d \|\nabla G_i^M\|^2_{\mathrm{L}^2(\gamma)}} }{\|\nabla_v \psi\|_{\mathrm{L}^2(\gamma)}} \right) ,
  \end{aligned}
  \end{equation}
  which concludes the proof.
\end{proof}

\begin{remark}\label{rmk34}
The choice of $G$ in the previous calculation is natural. Indeed, it yields an orthogonal decomposition in~$\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)$ of the term
\[
\left[(\partial_t + T) \, \Pi  h\right](t,x,v) =  (\partial_t \, \Pi \, h)(t,x) + G(v) \cdot  \int_{\mathcal{V}} G(v) \, (T \, \Pi \, h)(t,x,v') \gamma(dv').
\]
One component lies in the image of $\Pi$, while the second one is spanned by $G$ itself.    
Our result compares with \cite[Lemma 3.1]{albritton2019variational}, which is not quantitative at all, since the authors do not provide an explicit expression to terms like $M$ and $G$ in our proof.
We also compare with the estimates at the beginning of \cite[Theorem 2]{cao2019explicit}. In spite of the similarity in the strategy, we make the algebra behind the proof neat and usable also for general kinetic energies $\psi$. 
\end{remark}



\subsection{Proof of Theorem~\ref{thm1new}}
\label{sec:proof_thm1}

The exponential convergence is a direct consequence of \eqref{modpoi}. Indeed, note first that~\eqref{energy} implies
\begin{equation}\label{intpass}
\begin{aligned}
\frac{d \mathscr{H}_\tau(t)}{dt} &=\frac{d}{dt} \left( \int_0^\tau \|  h(t+s,\cdot,\cdot)\|^2_{\mathrm{L}^2(\Theta)} \, ds \right) =  \int_0^\tau \frac{d}{dt} \, \|  h(t+s,\cdot,\cdot)\|^2_{\mathrm{L}^2(\Theta)} \, ds   \\
&=-2\xi \int_0^\tau \| \nabla_v   h(t+s,\cdot,\cdot)\|^2_{\mathrm{L}^2(\Theta)} \, ds = -2 \xi \tau \int_0^\tau \| \nabla_v   h(t+s,\cdot,\cdot)\|^2_{\mathrm{L}^2(\Theta)} \, \mathrm{U}_\tau(ds).
\end{aligned}
\end{equation}
On the other hand, the space-time Poincar\'e inequality~\eqref{modpoi} implies, together with the bounds~\eqref{regest} and~\eqref{eq:coercivity_Delta_psi_OU}, that
\[
\bar{\lambda} \int_0^\tau \|    h(t+s,\cdot,\cdot)\|^2_{\mathrm{L}^2(\Theta)} \mathrm{U}_\tau(ds) \leq  \left(\frac{1}{c_\psi} + \xi^2\right) \int_0^\tau \| \nabla_v   h(t+s,\cdot,\cdot)\|^2_{\mathrm{L}^2(\Theta)} \mathrm{U}_\tau(ds).
\]
The combination of the previous inequality and \eqref{intpass} leads to
\[
\frac{d \mathscr{H}_\tau(t)}{dt} \leq -2  \bar{\lambda}\left(\frac{1}{\xi c_\psi} + \xi\right)^{-1}\mathscr{H}_\tau(t).
\]
from which  Theorem~\ref{thm1new} follows by a Gronwall inequality. 

\subsection{Proof of Corollary \ref{cor1new}}\label{subseccor1}

Decay estimates on~$\mathscr{H}_\tau(t)$ lead to pointwise in time decay estimates on~$h(t)$ since~$t \mapsto \|h(t)\|^2_{\mathrm{L}^2(\Theta)}$ is nonincreasing, which is a direct consequence of \eqref{energy}. Indeed, for any~$t \geq 0$,
\[
\| h(t+\tau) \|^2_{\mathrm{L}^2(\Theta)} \leq \frac1\tau \mathscr{H}_\tau(t) \leq \frac{\mathscr{H}_\tau(0)}{\tau} \mathrm{e}^{-2\lambda t} \leq \| h(0) \|^2_{\mathrm{L}^2(\Theta)} \mathrm{e}^{-2\lambda t},
\]
so that
\[
\forall t \geq 0, \qquad \| h(t) \|_{\mathrm{L}^2(\Theta)} \leq \| h(0) \|_{\mathrm{L}^2(\Theta)} \mathrm{e}^{-\lambda (t-\tau)},
\]
which gives the claimed result.

\section{Algebraic decay rates and Lyapunov functionals}\label{sec:sec4}

In this section we prove Theorem \ref{thm2new}, see Section \ref{sec:proof_thm2new}. Then, we discuss \eqref{moments} and related Lyapunov conditions in Section \ref{subsec:lyapunov}, specialising the analysis to cases of interest and proving Proposition \ref{prop1new} in Section \ref{subsec:prop1new}.    

\subsection{Proof of Theorem \ref{thm2new}}
\label{sec:proof_thm2new}

The overall strategy of the proof is similar to the one presented in~\cite{brigati2022time}, so that we only sketch the main steps of the argument. Note however that, compared to~\cite{brigati2022time} (and various other works, for instance~\cite{bouin2019hypocoercivity}), the transport part of the evolution now includes an additional term~$\nabla_x \phi \cdot \nabla_v$ and~$\nabla_v \psi \cdot \nabla_x$ replaces~$v \cdot \nabla_x$.

The overall idea for the proof is to rely on the Poincar\'e-type estimate~\eqref{modpoi}, where the second term is controlled by a combination of the weighted Poincar\'e inequality~\eqref{eq:weightedpoi} and a moment bound. 
More precisely, consider H\"older conjugate exponents, $p,q \geq 1,$ with $p = \sigma^{-1}(1+\sigma)>1.$ Note that $\frac{q}{p} = \sigma$.
Fix $t>0$, and define, for any~$s \in [0,\tau]$, the following moment based on~$\mathscr{G}$:
\begin{equation}\label{Zh}
Z_t(s) :=\iint_{\mathcal{X} \times\mathscr V} \mathscr G(v)^{\sigma} \,|h(t+s,x,v)-\Pi \, h(t+s,x)|^2\, \mu(dx)\,\gamma(dv).
\end{equation}
The first step of the proof is to bound~$\|(\mathrm{Id}- \Pi)h(t+\cdot,\cdot,\cdot)\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \Theta)}$. We start from the following pointwise bound for a given~$x \in  \mathcal{X}$, obtained with a H\"older inequality:
\[
\begin{aligned}
&\|h(t+s,x,\cdot)-\Pi \, h(t+s,x)\|_{\mathrm \mathrm{L}^2(\gamma)}^2 \\
&\qquad \leq \left(\int_{\mathscr V} \,\mathscr G(v)^{-1}\,|h(t+s,x,v)-\Pi \,h(t+s,x)|^2\,\gamma(dv) \right)^{\frac{1}{p}} \,  \\ 
& \qquad \quad\times\left( \int_{\mathscr V} \mathscr G(v)^{\frac{q}{p}}\,|h(t+s,x,v)-\Pi\,h(t+s,x)|^2\,\gamma(dv) \right)^{\frac{1}{q}}.
\end{aligned}
\]
Via \eqref{eq:weightedpoi}, the last right-hand side of the above inequality can be bounded as
$$P_\psi^{\frac{1}{p}} \left(\int_{\mathscr V}|\nabla_v h(t+s,x,v)|^2\,\gamma(dv) \right)^{\frac{1}{p}} \, \left( \int_{\mathscr V} \mathscr G(v)^{\frac{q}{p}}\,|h(t+s,x,v)-\Pi\,h(t+s,x)|^2\,\gamma(dv) \right)^{\frac{1}{q}} .$$
Integrating with respect to the measure $\mathrm{U}_\tau \otimes \mu,$ and applying H\"older's inequality in the variables $s,x$ gives
\[
\begin{aligned}
&\int_0^\tau \|h(t+s,\cdot,\cdot)-\Pi \, h(t+s,\cdot)\|_{\mathrm \mathrm{L}^2(\Theta)}^2 \,\mathrm{U}_\tau(ds)\\
&\qquad \leq P_\psi^{\frac{1}{p}} \,\left( \int_0^\tau \, \int_{\mathscr X}  \,\int_{\mathscr V}|\nabla_v h(t+s,x,v)|^2\,\mathrm{U}_\tau(ds) \, \Theta(dx \, dv) \right)^{\frac{1}{p}} \\ 
&\qquad\quad \times\left( \int_0^\tau \, \int_{\mathscr X} \,\int_{\mathscr V} \mathscr G(v)^{\frac{q}{p}}\,|h(t+s,x,v)-\Pi\,h(t+s,x)|^2\,\mathrm{U}_\tau(ds) \, \Theta(dx \, dv) \right)^{\frac{1}{q}} \\ 
&\qquad= P_\psi^{\frac{1}{p}} \,\left( \int_0^\tau \, \|\nabla_v h(t+s,\cdot,\cdot)\|_{\mathrm \mathrm{L}^2(\Theta)}^2\,\mathrm{U}_\tau(ds) \,  \right)^{\frac{1}{p}} \, \left( \int_0^\tau Z_t(s) \,\mathrm{U}_\tau(ds) \right)^{\frac{1}{q}} \\
&\qquad\leq P_\psi^{\frac{1}{p}} \,\left( \int_0^\tau \, \|\nabla_v h(t+s,\cdot,\cdot)\|_{\mathrm \mathrm{L}^2(\Theta)}^2\,\mathrm{U}_\tau(ds) \,  \right)^{\frac{1}{p}} \,  W^{\frac{1}{q}} \left( \|h_0\|^2_{\mathrm{L}^2(\Theta)} + \mathscr{U}(h_0) \right)^{\frac{1}{q}},
\end{aligned}
\]
where the last inequality follows from Lemma \ref{lem4} below. Combining the last bound with~\eqref{modpoi}, and then using~\eqref{regest} gives 
\[
\begin{aligned}
 &\bar{\lambda} \int_0^\tau \|h(t+s,\cdot,\cdot)\|^2_{\mathrm{L}^2( \Theta)} \, \mathrm{U}_\tau(ds) \\
 & \qquad \leq \int_0^\tau \|(\partial_t + T) \,h(t,\cdot,\cdot)\|^2_{\mathrm{L}^2(\mu;\mathrm{H}^{-1}(\gamma))}  \\
 &\qquad \quad+ W^{\frac{1}{q}} \left( \|h_0\|^2_{\mathrm{L}^2(\Theta)}+ \mathscr{U}(h_0) \right)^{\frac{1}{q}} \, P_\psi^{\frac{1}{p}} \, \left( \int_0^\tau \|\nabla_v h(t+s,\cdot,\cdot)\|^{2}_{\mathrm{L}^2( \Theta)} \, \mathrm{U}_\tau(ds) \right)^{\frac{1}{p}} \\
&\qquad \leq \xi^2 \, \int_0^\tau \|\nabla_v h(t+s,\cdot,\cdot)\|^2_{\mathrm{L}^2( \Theta)} \, \mathrm{U}_\tau(ds) \\ 
&\qquad \quad + W^{\frac{1}{q}} \left( \|h_0\|^2_{\mathrm{L}^2(\Theta)}+ \mathscr{U}(h_0) \right)^{\frac{1}{q}} \, P_\psi^{\frac{1}{p}} \, \left( \int_0^\tau \|\nabla_v h(t+s,\cdot,\cdot)\|^{2}_{\mathrm{L}^2( \Theta)} \, \mathrm{U}_\tau(ds) \right)^{\frac{1}{p}}.
\end{aligned}
\]
In view of the definitions \eqref{timeavg} and \eqref{intpass}, the latter inequality can be rewritten as 
\begin{equation}\label{diffeq}
\mathscr{H}_\tau(t) \leq \frac{\xi}{2 \, \bar{\lambda}} \,  \, \left(- \frac{d}{dt} \, \mathscr{H}_\tau(t)\right) +   \xi^{-\frac{1}{p}} \,  \, M_0 \, \left(-\frac{d}{dt} \, \mathscr{H}_\tau(t) \right)^{\frac{1}{p}},
\end{equation} 
where $$M_0 := \tau^{\frac{1}{\sigma+1}} \, \frac{1}{\bar{\lambda}}  \, \left( \frac{P_\psi}{2} \right)^{\frac{1}{p}}\, W^{\frac{1}{q}} \left( \|h_0\|^2_{\mathrm{L}^2(\Theta)} + \mathscr{U}(h_0) \right)^{\frac{1}{q}} .$$

The proof can now be concluded using classical arguments, relying on a generalization of the Gronwall inequality leading to algebraic decay rates. We however carefully keep track of the parameters influencing the convergence rate, by retaining the largest term in~\eqref{diffeq}. In order to make this precise, consider $$y \mapsto \vartheta(y) = \, \frac{\xi}{2\bar{\lambda}} \, y  + M_0 \left( \frac{y}{\xi} \right)^{\frac{1}{p}},$$
which is strictly increasing from $[0,\infty)$ to $[0,\infty).$
Set $y_0 = \vartheta^{-1}(\mathscr{H}_\tau(0)).$
Note that 
$$\forall y \leq y_0, \quad \vartheta(y) \leq \left( \xi^{-\frac{1}{p}} M_0 +  \, \frac{\xi}{2\bar{\lambda}} \, y_0^{\frac{p-1}{p}} \right) \, y^{\frac{1}{p}}.$$
Hence, as $t \mapsto \mathscr{H}_\tau(t)$ is nonincreasing, we have
$$ \mathscr{H}_\tau(t) \leq \left( \xi^{-\frac{1}{p}} M_0 +  \,  \frac{\xi}{2\bar{\lambda}} \, y_0^{\frac{p-1}{p}} \right) \, \left( - \frac{d}{dt} \, \mathscr{H}_\tau(t)\right)^{\frac{1}{p}}.$$
Applying the standard Bihari--LaSalle \cite{bihari1956generalization,lasalle1949uniqueness} argument, one proves, as in \cite{brigati2022time},
\begin{equation}\label{blsimplified}
\mathscr{H}_\tau(t) \leq \left( \mathscr{H}_\tau(0)^{1-p} + (p-1) \left( \xi^{-\frac{1}{p}} M_0 +  \frac{\xi}{2\bar{\lambda}} \, y_0^{\frac{p-1}{p}} \right)^{-p} t \right)^{-\frac{1}{p-1}}.
\end{equation}
Equation \eqref{blsimplified} can be rewritten as 
$$ \mathscr{H}_\tau(t) \leq \frac{\mathscr{H}_\tau(0)}{\left( 1 + \left( \xi^{-\frac{1}{p}} c_1 + \xi c_2 \right)^{-p} t \right)^{\frac{1}{p-1}}},$$
with $c_1 = (p-1)^{- \frac 1p} \, M_0 \mathscr{H}_\tau(0)^{p \, (1-p)}$ and $c_2 = (p-1)^{- \frac 1p } \, \frac{\xi}{2\bar{\lambda}} \, y_0^{\frac{p-1}{p}} \, \mathscr{H}_\tau(0)^{\frac{1-p}{p}}.$
The proof of Theorem \ref{thm2new} is concluded.

\begin{remark}
A more accurate estimate could be achieved by rewriting \eqref{diffeq} as
$$\frac{d}{dt} \, \mathscr{H}_\tau(t) \leq - \zeta(\mathscr{H}_\tau(t)),$$
with $\zeta = \theta^{-1},$ which is indeed invertible, and then integrating the latter inequality.
\end{remark}

\subsection{Evolution of moments and Lyapunov conditions}\label{subsec:lyapunov} 

The proof of Theorem~\ref{thm2new} written in Section~\ref{sec:proof_thm2new} relies on moment estimates encoded by~$Z_t(s)$ introduced in~\eqref{Zh} (see Lemma \ref{lem4} below). Bounds on this quantity can be obtained by introducing a suitable Lyapunov function, which we now make precise.  

Let $U:\mathbb{R}^{2d} \to [1,\infty)$ be a continuous function, and fix $\eta \geq 0$. Define, for any measurable function $g$, the functional 
\begin{equation}\label{eq:lyapunov}
    \mathscr{U}_\eta(g) := \int_{\mathscr X \times \mathscr V} U(x,v)^\eta g^2(x,v) \, \Theta(dx \, dv). 
\end{equation}
In order to establish \eqref{moments}, our goal is to find a function $U$ and a constant $A \in \mathbb R_+$ such that $\mathscr{U}_\eta(h(t,\cdot,\cdot)) \leq A \, \mathscr{U}_\eta(h_0)$ for any $h$ solving \eqref{VOU}.
We first motivate that it suffices to this end to establish a suitable Lyapunov condition, given in Lemma \ref{lyapulem} below, which is inspired by \cite[Proposition 4]{bouin2019hypocoercivity}.

\begin{lemma}\label{lyapulem}
Consider $\sigma>0$, defined in Theorem \ref{thm2new}. Assume that there exist $\rho,\eta > 0$ such that $U^{\eta} \geq \rho \, \mathscr G^{\sigma},$ and that, for any $\eta' \geq \eta$, there exist four numbers $\ell_{\eta'},a_{\eta'},b_{\eta'}, R_{\eta'} > 0$ (which depend on $\eta'$) such that 
  \begin{equation}\label{eq:lyapunovabstr}
      \left((\xi \mathscr{S} - T ) \, U^{\eta'}\right) \,(x,v) \leq - \xi b_{\eta'} \, U(x,v)^{\eta' - \ell_{\eta'}} + \xi a_{\eta'} \mathbb{1}_{B(0,R_{\eta'})}(x,v).  
\end{equation} 
Then, for any $\eta' \geq \eta$ such that $\eta' > \eta + 2 \ell_{\eta'}$, and for any solution~$h$ to \eqref{VOU}, it holds 
$$\mathscr{U}_\eta(h(t,\cdot,\cdot)) \leq \, A_{\eta,\eta'} \, \mathscr U_\eta(h_0),$$
for a constant $A_{\eta,\eta'} \in {\mathbb R}_+,$ which can be uniformly bounded with respect to $\xi.$ 
\end{lemma}
An immediate corollary of this result is that~\eqref{moments} holds for $\mathscr{U} = \mathscr{U}_{\eta},$ and $A = A_{\eta,\eta'}$. The proof of Lemma \ref{lyapulem} can be read in Appendix \ref{app:C}, where an explicit expression for~$A$ can be found. We next show how Lemma~\ref{lyapulem} allows to obtain the control of the moment $Z(t)$ defined in~\eqref{Zh}, which appears in the proof of Theorem~\ref{thm2new}.

\begin{lemma}\label{lem4}
There is an explicit constant $W>0$ (not depending on $d,\xi$), such that, for any solution $h \in \mathrm{H}_{\mathrm{kin}}$ to~\eqref{VOU} with initial datum $h_0$ with average zero with respect to $\Theta$ and such that \eqref{moments} holds,
$$\forall\,t\ge0, \qquad Z(t)\le W \left( \|h_0\|^2_{\mathrm{L}^2(\Theta)} + \mathscr{U}(h_0) \right).$$
\end{lemma}

\begin{proof}
We bound $Z(t)$ as 
$$Z(t) \leq 2 \iint h(t,x,v)^2 \, \mathscr G(v)^\sigma \, \Theta(dx,dv) + 2\iint (\Pi h)^2(t,x) \, \mathscr G(v)^\sigma \, \Theta(dx \,dv).$$ 
The first term in the right-hand side of the previous inequality is bounded with \eqref{moments} as $$\iint h(t,x,v)^2 \mathscr G(v)^\sigma \, \Theta(dx \, dv) \leq A \, \mathscr{U}(h_0).$$
On the other hand, 
\[
\begin{aligned}
 \iint_{\mathscr{X} \times \mathscr{V}} (\Pi h)^2(t,x)  \mathscr G(v)^\sigma \, \Theta(dx\,dv) & = \int_{\mathscr X} (\Pi h)^2(t,x) \mathscr G(v)^\sigma \, \mu(dx) \, \gamma(v) \\
 &\leq \|h_0\|^2_{\mathrm{L}^2(\Theta)} \, \left\|\mathscr G(v)^\sigma\right\|_{\mathrm{L}^1(d\gamma)},
\end{aligned}
\]
where we used a Cauchy--Schwarz inequality in the last step to write $\|\Pi h(t,\cdot)\|^2_{\mathrm{L}^2(\mu)} \leq \|h(t,\cdot,\cdot)\|^2_{\mathrm{L}^2(\Theta)} \leq \|h_0\|^2_{\mathrm{L}^2(\Theta)}$.
The result therefore follows with $W = 2 \max(A, \left\|\mathscr G(v)^\sigma\right\|_{\mathrm{L}^1(d\gamma)}).$
\end{proof}

\subsection{Proof of Proposition \ref{prop1new}}\label{subsec:prop1new} 

It suffices at this stage to check that Lemma \ref{lyapulem} can be applied, and in particular to prove~\eqref{eq:lyapunovabstr} by distinguishing two situations: sub-exponential local equilibria in Section \ref{sss:se}, and fat-tailed local equilibria in Section \ref{sss:fat}.  

\subsubsection{Sub-exponential local equilibria}\label{sss:se}

\begin{proof}[Proof of Proposition \ref{prop1new} -- Part (1)]
The existence of $\rho \in \mathbb R_+$ in Lemma \ref{lyapulem} is guaranteed by~\eqref{subexponential1}.
Let $U(x,v) = \phi(x)+\psi(v)$, and define $\mathscr U_\eta$ as in \eqref{eq:lyapunov}.
Since $T U^\eta = 0,$  it suffices in view of \eqref{eq:lyapunovabstr} to consider
\[
\begin{aligned}
\mathscr{S} (\phi + \psi)^\eta &=  -\eta (\phi+\psi)^{\eta-1} |\nabla_v \psi|^2 + \eta \, \mathrm{div}_v \left( (\phi+\psi)^{\eta-1} \nabla_v \psi \right) \\
&= - \eta \, (\phi+\psi)^{\eta-1} \left[ \left(1- \frac{\eta-1}{\phi+\psi} \right) |\nabla_v \psi|^2 - \Delta_v \psi  \right].
\end{aligned} 
\]
Note that
$$\liminf_{|x|,|v| \to \infty} \frac{\mathscr S \, (\phi+\psi)^{\eta}}{(\phi+\psi)^{\eta-\ell}} =  - \eta \, (\phi+\psi)^{\ell-1} \left[ \left(1- \frac{\eta-1}{\phi+\psi} \right) |\nabla_v \psi|^2 - \Delta_v \psi  \right] \leq - \delta \eta < 0,$$
for some $\delta >0$, whose existence is ensured by \eqref{subexponential2}.
The last bound implies \eqref{eq:lyapunovabstr}, with  $b_{\eta'} \leq \delta \eta$ arbitrarily close to $\delta \,\eta$ (upon increasing $a_{\eta'}$ and $R_{\eta'})$.
\end{proof}


A typical situation for which condition \eqref{subexponential2} is satisfied is when $\psi(v)$ is proportional to $(1+|v|^2)^{\frac{\alpha}{2}},$ for $\alpha \in (0,1),$ see \eqref{exmp1}. One can check that Proposition \ref{prop1new} -- (1) holds for $\ell \geq \frac{2}{\alpha} - 1$. In \eqref{eq:weightedpoi}, one can consider $\mathscr G(v)=\left(1 + |v|^2\right)^{1-\alpha}.$ 
Hence, in order to satisfy hypothesis \eqref{subexponential1} of Proposition \ref{prop1new}, we also require
$$\eta > \sigma \frac{2-2\alpha}{\alpha}.$$


\subsubsection{Fat tails local equilibria}\label{sss:fat}
When $\psi$ is weaker than polynomial at infinity, the estimates of Section~\ref{sss:se} are of no help, because they degenerate as $\alpha \to 0^+.$ The function $U$ has to be modified accordingly.


\begin{proof}[Proof of Proposition \ref{prop1new} -- Part (2)]
Note first that~\eqref{fattail2} is sufficient to ensure the existence of $\rho \in \mathbb R_+$ in Lemma \ref{lyapulem}.
Let $U = \mathrm{e}^{\phi+\psi}.$ 
Since $T U^\eta =0,$ it suffices again to calculate only $\xi \mathscr{S}\left(U^\eta\right)$. We write 
\[
\limsup_{|x|,|v| \to \infty} \mathscr{S}\left(U^\eta\right) =  \mathscr{S} \left( \mathrm{e}^{\eta(\phi+\psi)} \right) = - \eta \, \mathrm{e}^{\eta(\phi+\psi)} \left[ (1-\eta) |\nabla_v \psi|^2 - \Delta_v \psi \right],
\]
so that
$$\limsup_{|x|,|v|\to \infty}\frac{\mathscr S \left(U^\eta\right) }{U^{\eta-\ell}} = \limsup_{|x|,|v|\to \infty} -\eta \, U^{\ell} \,  \left[ (1-\eta) |\nabla_v \psi|^2 - \Delta_v \psi \right] \leq - \delta \eta,$$
for some $\delta >0,$ in view of \eqref{fattail1}.  
This shows that \eqref{eq:lyapunovabstr} holds. 
\end{proof}

Let us now discuss kinetic energy function satisfying \eqref{fattail2}-\eqref{fattail1}. Assume that $d \geq 3$ and $\psi(v)$ is of the form $\beta \log \sqrt{1+|v|^2},$ for some $\beta > d+2,$ so that $\mathrm{e}^{-\psi(v)}$ is integrable and Assumption \ref{ass2} holds, see \eqref{exmp2}. Then, 
$$\nabla_v \psi(v) =  \, \beta \, \frac{v}{ 1+|v|^2}, \qquad \Delta_v \psi(v) = \beta \frac{d + (d-2) |v|^2 }{(1+|v|^2)^2}.$$ 
For $\frac{2}{\beta} < \eta < 1$, condition \eqref{fattail1} in Proposition \ref{prop1new} holds true with
$$\ell = \frac{2}{\beta}.$$ 
If, in addition, $\sigma < \frac{\eta \beta}{2},$ then \eqref{fattail2}, hence all hypotheses of Lemma \ref{lyapulem}, are satisfied. 




\section{Weighted Poincar\'e-Lions inequalities}\label{sec:sec5}

We present a constructive proof, based on a generalization of the manipulations performed in~\cite[Lemma~2.6]{cao2019explicit}. We extend the approach to the situation when the operator~$\nabla_x^* \nabla_x$ has a spectral gap but not necessarily a discrete spectrum. We also present the argument in a more abstract manner, in order to highlight the key steps in the construction.

\subsection{Reduction to solving a divergence equation}

Recall the space $\mathrm{H}^1_\mathrm{DC}(\mathrm{U}_\tau \otimes \mu)$, defined in \eqref{dcdef}.
The Lions inequality~\eqref{lionstx} is implied by the following property (in fact, \cite{amrouche2015lemma} shows the equivalence between the two statements): there exists a positive constant $C_\tau'$ such that, for any~$f \in \mathrm{L}_0^2(\mathrm{U}_\tau \otimes \mu)$, there exists a solution~$Z = (Z_{0},Z_{1},\dots,Z_d) \in \mathrm{H}^1_\mathrm{DC}(\mathrm{U}_\tau \otimes \mu)^{d+1}$ to
\begin{equation}
  \label{eq:div_eq_Z}
  -\partial_t Z_0 + \sum_{i=1}^d \partial_{x_i}^* Z_i = f,
\end{equation}
which is such that 
\begin{equation}
  \label{elliptic}
  \|Z\|_{\mathrm{H}^1(\mathrm{U}_\tau \otimes \mu)} = \sqrt{\sum_{i=0}^d \|Z_i \|^2_{\mathrm{H}^1(\mathrm{U}_\tau \otimes \mu)}} \leq C'_\tau \, \|f\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)}. 
\end{equation}
Of course, there may be many solutions to the divergence-like equation~\eqref{eq:div_eq_Z}. When such a solution exists, 
\[
\begin{aligned}
  \|f\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)}
  & = \int_0^\tau \int_\mathcal{X} f(t,x)^2 \, \mathrm{U}_\tau(dt) \, \mu(dx) = \int_0^\tau \int_\mathcal{X} \left(-\partial_t Z + \sum_{i=1}^d \partial_{x_i}^* Z\right) \, f \, d\mathrm{U}_\tau \, d\mu \\
  & = \left\langle \nabla_{t,x} f, Z\right\rangle_{\mathrm{H}^{-1}(\mathrm{U}_\tau \otimes \mu), H^{1}_\mathrm{DC}(\mathrm{U}_\tau \otimes \mu)} \leq \|Z\|_{\mathrm{H}^1(\mathrm{U}_\tau \otimes \mu)} \|\nabla_{t,x} f\|_{\mathrm{H}^{-1}(\mathrm{U}_\tau \otimes \mu)} \\
  & \leq C'_\tau \|f\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)} \|\nabla_{t,x} f\|_{\mathrm{H}^{-1}(\mathrm{U}_\tau \otimes \mu)}.
\end{aligned}
\]
Hence, Lions' inequality \eqref{lionstx} is recovered with~$C^{\mathrm{Lions}}_\tau \leq (C'_\tau)^2$. We therefore concentrate in this section on obtaining solutions to~\eqref{eq:div_eq_Z} which satisfy~\eqref{elliptic}.   

\subsection{Solving the divergence equation}

We show in this section how to solve~\eqref{eq:div_eq_Z} in order to have estimates such as~\eqref{elliptic}. The precise result is the following.


\begin{theorem}
  Suppose that \eqref{poiphi} and \eqref{hess} hold. Then, there exists an explicit~$C_\phi \in \mathbb{R}_+$, depending only on $c_\phi,c'_\phi,c''_\phi$ and not explicitly on $d$, such that, for any~$\tau > 0$ and~$f \in \mathrm{L}^2_0(\mathrm{U}_\tau \otimes \mu)$, the equation posed on the space-time domain~$[0,\tau] \times \mathcal{X}$
  \begin{equation}
    \label{eq:Lions_A}
    -\partial_t Z_0 + \sum_{i=1}^d \partial_{x_i}^* Z_i = f,
  \end{equation}
  admits a solution~$(Z_0,\dots,Z_d) \in \mathrm{H}^1_{\mathrm{DC}}(\mathrm{U}_\tau \otimes \mu)^{d+1}$ which satisfies
  \begin{equation}
    \label{eq:estimates_solution_divergence_equation}
    \sum_{i=0}^d \| Z_i \|^2_{\mathrm{H}^1(\mathrm{U}_\tau \otimes \mu)} \leq C_\phi \, \left(1 + \sqrt{d}\right) \, \left(\frac{1}{\tau^2}+\tau^2\right)^2 \, \|f\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)}^2.
  \end{equation}
\end{theorem}
In fact, considering the positive self-adjoint operator satisfying (in view of \eqref{poiphi}) the following coercivity inequality on~$\mathrm{L}^2_0(\mu)$: 
\[
\mathcal{L} = \nabla_x^* \nabla_x \geq c_\phi,
\]
and its square-root denoted by $L = \mathcal{L}^{1/2}$, we construct an explicit solution as follows by generalising the construction in \cite{cao2019explicit}: 
\begin{equation}
  \label{eq:explicit_solution_divergence_equation}
  \phi = \nabla_{t,x} \mathscr{W}^{-1} P_{\mathcal{N}^\perp} f + \begin{pmatrix} F_0(t,\mathcal{L}) \\ \partial_{x_1} F_1(t,\mathcal{L}) \\ \vdots \\ \partial_{x_d} F_1(t,\mathcal{L}) \end{pmatrix} P_{\mathcal{N},+} f + \begin{pmatrix} F_0(\tau-t,\mathcal{L}) \\ \partial_{x_1} F_1(\tau-t,\mathcal{L}) \\ \vdots \\ \partial_{x_d} F_1(\tau-t,\mathcal{L}) \end{pmatrix} P_{\mathcal{N},-} f,
\end{equation}
where we introduced the following objects:
\begin{itemize}
  \item the operator~$P_{\mathcal{N}^\perp}$ is the projector onto the orthogonal of the vector space~$\mathcal{N} \subset \mathrm{L}^2_0(\mathrm{U}_\tau \otimes \mu)$ composed of linear combinations of forward and backward wave-like functions: $g \in \mathcal{N}$ if and only if there exist~$g_+,g_- \in \mathrm{L}^2_0(\mu)$ such that 
\begin{equation}
  \label{eq:decomposition_elements_N}
  g(t,x) = \left(\mathrm{e}^{-t L}g_+\right)(x) + \left(\mathrm{e}^{-(\tau-t) L} g_-\right)(x); 
\end{equation}
\item the operator~$\mathcal{W} = -\partial_t^2 + \mathcal{L}$ is considered on~$\mathrm{L}^2_0(\mathrm{U}_\tau \otimes \mu)$ and endowed with Neumann boundary conditions in time; 
\item the bounded operators~$F_0(t,\mathcal{L})$ and~$F_1(t,\mathcal{L})$ respectively act as
  \[
\begin{aligned}
  F_0(t,\mathcal{L}) & = -L^{-1} \left(1+\mathrm{e}^{-\tau L}\right)\left(1-\mathrm{e}^{-\tau L}\right)^{-2} \left(1-\mathrm{e}^{-t L}\right)\left(1-\mathrm{e}^{-(\tau-t) L}\right)\left(\mathrm{e}^{-t L} - 2\mathrm{e}^{-\tau L}(1+\mathrm{e}^{-\tau L})^{-1} \right), \\
  F_1(t,\mathcal{L}) & = 3L^{-2} \left(1+\mathrm{e}^{-\tau L}\right)\left(1-\mathrm{e}^{-\tau L}\right)^{-2} \left(1-\mathrm{e}^{-t L}\right)\left(1-\mathrm{e}^{-(\tau-t) L}\right);
\end{aligned}
\]
\item the projectors~$P_+,P_- : \mathrm{L}^2_0(\mathrm{U}_\tau \otimes \mu) \to \mathcal{N} \subset \mathrm{L}^2_0(\mu)$ are defined as
\begin{equation}
  \label{eq:def_Q_pm}
  \begin{aligned}
    \left(P_{\mathcal{N},+} f\right)(t) & = \mathrm{e}^{-t L} \left(\int_0^\tau G_+(s,L) \mathrm{e}^{-s L} \, \mathrm{U}_\tau(ds) \right)^{-1} \int_0^\tau G_+(s,L)f(s) \, \mathrm{U}_\tau(ds), \\
    \left(P_{\mathcal{N},-} f\right)(t) & = \mathrm{e}^{-(\tau-t) L} \left(\int_0^\tau G_-(s,L) \mathrm{e}^{-(\tau-s)L} \, \mathrm{U}_\tau(ds) \right)^{-1} \int_0^\tau G_-(s,L)f(s) \, \mathrm{U}_\tau(ds), 
\end{aligned}
\end{equation}
with
\begin{equation}
  \label{eq:U_tau_etc}
G_+(s,L) = U_\tau^{-1} \mathrm{e}^{-(2\tau-s) L}-\mathrm{e}^{-s L},
\qquad
G_-(s,L) = U_\tau^{-1} \mathrm{e}^{-(\tau+s) L}-\mathrm{e}^{-(\tau-s) L},
\qquad
U_\tau = \frac1\tau \int_0^\tau \mathrm{e}^{-2 s L} \, ds.
\end{equation}
\end{itemize}
The form of this solution is motivated in Section~\ref{sec:heuristic_construction}, while the estimates~\eqref{eq:estimates_solution_divergence_equation} are proved in Section~\ref{sec:proof_estimates_divergence}. Note already that~$P_{\mathcal{N},+}$ and~$P_{\mathcal{N},-}$ are bounded operators since~$P_{\mathcal{N},+}^2 = P_{\mathcal{N},+}$ and~$P_{\mathcal{N},-}^2 = P_{\mathcal{N},-}$. To obtain the estimates~\eqref{eq:estimates_solution_divergence_equation} for the last two terms in~\eqref{eq:explicit_solution_divergence_equation}, it is therefore sufficient to obtain uniform-in-time bounds on the operators~$F_0(t,\mathcal{L})$, $\partial_t F_0(t,\mathcal{L})$, $\partial_{x_i} F_0(t,\mathcal{L})$ and~$\partial_{x_i} F_1(t,\mathcal{L}),\partial_t \partial_{x_i} F_1(t,\mathcal{L}),\partial_{x_i,x_j}^2 F_1(t,\mathcal{L})$ considered on~$\mathrm{L}^2(\mu)$.

An inspection of the proof finally shows that the factor~$\tau^2$ on the right hand side of~\eqref{eq:estimates_solution_divergence_equation} comes from the first part of the solution~\eqref{eq:explicit_solution_divergence_equation} (it can be traced back to some coercivity estimate based on the Poincaré inequality for~$\mathrm{U}_\tau$), while the factor~$\tau^{-2}$ arises from the part of the solution arising from~$P_\mathcal{N} f$, because of the operator~$\left(1-\mathrm{e}^{-\tau L}\right)^{-2}$ in the expressions of~$F_0(t,\mathcal{L})$ and~$F_1(t,\mathcal{L})$.

\subsection{Heuristic construction of the solution}
\label{sec:heuristic_construction}

To find one possible solution to~\eqref{eq:Lions_A}, a natural idea is to look for~$(Z_0,\dots,Z_d)$ of the form~$(\partial_t u, \partial_{x_1} u, \dots,\partial_{x_d} u)$, with~$u$ the unique solution of the following equation
\begin{equation}
  \label{eq:eq_to_solve_with_LaxMilgram}
  (-\partial_t^2 + \mathcal{L})u = f, \qquad \partial_t u(0,\cdot) = \partial_t u(\tau,\cdot) = 0.
\end{equation}
The functions~$\partial_{x_i} u(0,\cdot)$ and~$\partial_{x_i} u(\tau,\cdot)$ are however not~0 in general. They turn out to vanish provided the right hand side of~\eqref{eq:Lions_A} is restricted to the orthogonal of the vector space~$\mathcal{N}$. Note that functions~$g \in \mathcal{N}$ satisfy~$(-\partial_t^2 + \mathcal{L})g = 0$ in the weak sense. This equality is satisfied in the strong sense when~$g_+,g_-$ are sufficiently regular, namely~$g_+,g_- \in D(\mathcal{L})$. The first step in the proof, as done in~\cite[Lemma~2.4]{cao2019explicit}, is to solve~\eqref{eq:Lions_A} for the part of the right hand side~$f$ which is in~$\mathcal{N}^\perp$, using~\eqref{eq:eq_to_solve_with_LaxMilgram} and the \textit{Lax--Milgram} lemma see~\cite[Chapter 5]{brezis2010functional} for instance.




\begin{lemma}
  \label{lem:Lax_Milgram}
  Suppose \eqref{hess} and \eqref{poiphi}.
  Fix~$\tau > 0$. For any~$f \in \mathrm{L}^2_0(\mathrm{U}_\tau \otimes \mu)$, there exists a unique weak solution~$u \in H^2(\mathrm{U}_\tau \otimes \mu) \cap \mathrm{L}^2_0(\mathrm{U}_\tau \otimes \mu)$ to~\eqref{eq:eq_to_solve_with_LaxMilgram}: for any~$w \in \mathrm{H}^1(\mathrm{U}_\tau \otimes \mu)$,
  \begin{equation}
    \label{eq:weak_formulation_LM}
  \int_0^\tau \int_\mathcal{X} \left[ (\partial_t u)(\partial_t w) + \nabla u \cdot \nabla w \right]\, \mathrm{U}_\tau(dt) \, \mu(dx) =\int_0^\tau \int_\mathcal{X} f w \, \mathrm{U}_\tau(dt) \,\mu(dx). 
  \end{equation}
  Moreover, 
  \[
  \|u \|_{H^2(\mathrm{U}_\tau \otimes \mu)} \leq \left(C^2_{\mathrm P} + C^4_{\mathrm{P}} + 2 c'_\phi \left(\sqrt{d} + 2 \max \left(8 c'_\phi , \sqrt{c''_\phi \,d} \right)  \,  \right)\right)^{\frac{1}{2}} \, \|f\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)},
  \]
  with $C_{\mathrm P} = \min \left( c_\phi, \frac{\pi}{\tau}\right).$
\end{lemma}


The second step is to construct a solution for the part of the right hand side in~$\mathcal{N}$. This is done in two steps: (i) providing the expression for the orthogonal projection~$P_\mathcal{N} = P_{\mathcal{N},+} + P_{\mathcal{N},-}$ of~$\mathrm{L}^2_0(\mathrm{U}_\tau \otimes \mu)$ onto~$\mathcal{N}$; and then (ii) solving the divergence equation~\eqref{eq:div_eq_Z} with right hand side~$P_{\mathcal{N},\pm} f$. To identify~$P_{\mathcal{N},\pm}$, we write 
\begin{equation}
  \label{eq:decomposition_P_N}
  (P_\mathcal{N} f)(t,x) = \left(\mathrm{e}^{-t L} Q_+ f\right)(x) + \left(\mathrm{e}^{-(\tau-t) L} Q_- f\right)(x).
\end{equation}
By definition of the orthogonal projection, it holds, for any~$w_+,w_- \in \mathrm{L}^2(\mu)$, 
\[
\begin{aligned}
  \int_0^\tau \int_\mathcal{X} \left[f(t,x)-(P_\mathcal{N} f)(t,x)\right]\left(\mathrm{e}^{-t L} w_+\right)(x) \, \mathrm{U}_\tau(dt) \, \mu(dx) & = 0, \\
  \int_0^\tau \int_\mathcal{X} \left[f(t,x)-(P_\mathcal{N} f)(t,x)\right]\left(\mathrm{e}^{-(\tau-t) L} w_-\right)(x) \, \mathrm{U}_\tau(dt) \, \mu(dx) & = 0.
  \end{aligned}
\]
Therefore, using the self-adjointness of $\mathrm{L},$
\begin{equation}
  \label{eq:system_for_projection}
  \begin{aligned}
    \int_\mathcal{X} (U_\tau Q_+ f) w_+ \, d\mu + \int_\mathcal{X} \left(\mathrm{e}^{-\tau L} Q_- f\right) w_+ \, d\mu & = \int_\mathcal{X} \left[\int_0^\tau \mathrm{e}^{-t L} f(t) \, \mathrm{U}_\tau(dt) \right] w_+ \, d\mu, \\
    \int_\mathcal{X} \left(\mathrm{e}^{-\tau L} Q_+ f\right) w_- \, d\mu + \int_\mathcal{X} (U_\tau Q_- f) w_- \, d\mu& = \int_\mathcal{X} \left[\int_0^\tau \mathrm{e}^{-(\tau-t) L} f(t) \, \mathrm{U}_\tau(dt) \right] w_- \, d\mu, \\ 
  \end{aligned}
\end{equation}
where we recall from~\eqref{eq:U_tau_etc} that
\[
U_\tau = \int_0^\tau \mathrm{e}^{-2t L} \, \mathrm{U}_\tau(dt) = \frac{1-\mathrm{e}^{-2\tau L}}{2\tau} L^{-1}.
\]
Choosing~$w_+ = U_\tau^{-1} \mathrm{e}^{-\tau L} w_-$ and subtracting the second equation of~\eqref{eq:system_for_projection} from the first one leads to
\[
\int_\mathcal{X} \left[\left(U_\tau^{-1}\mathrm{e}^{-2\tau \mathrm{L}^{1/2}} - U_\tau\right) Q_- f\right] w_- \, d\mu = \int_\mathcal{X} \left[\int_0^\tau \left(U_\tau^{-1} \mathrm{e}^{-(\tau+t) \mathrm{L}^{1/2}}-\mathrm{e}^{-(\tau-t)\mathrm{L}^{1/2}}\right) f(t) \, \mathrm{U}_\tau(dt) \right] w_- \, d\mu.
\]
Since the latter equality holds for any $w_ \in \mathrm{L}^2(\mu),$ it follows that   
\[
Q_- f = \left(U_\tau^{-1}\mathrm{e}^{-2\tau \mathrm{L}^{1/2}} - U_\tau\right)^{-1} \int_0^\tau \left(U_\tau^{-1} \mathrm{e}^{-(\tau+t) \mathrm{L}^{1/2}}-\mathrm{e}^{-(\tau-t) \mathrm{L}^{1/2}}\right) f(t) \, \mathrm{U}_\tau(dt),
\]
which leads to~\eqref{eq:def_Q_pm} in view of~\eqref{eq:decomposition_P_N}. The formula for~$Q_+ f$ is obtained in a similar manner.
The operators $Q_{\pm}$ are well-defined, since the pre-factor of the integrals reads 
$$\left(U_\tau^{-1}\mathrm{e}^{-2\tau \mathrm{L}^{1/2}} - U_\tau\right)^{-1} \, U_\tau^{-1} = \mathrm{L}^2 \left(\mathrm{L}^2 \mathrm{e}^{-2\tau L} - \left( 1 - \mathrm{e}^{-2\tau L} \right)^2\right)^{-1}.$$
The latter operator is well defined since $\mathrm{L}^2 \mathrm{e}^{-2\tau L} - \left( 1 - \mathrm{e}^{-2\tau L} \right)^2$ is coercive on $\mathrm{L}^2(\mu),$ hence invertible.

Let us next solve the divergence equation~\eqref{eq:div_eq_Z} with right hand side~$(P_{\mathcal{N},+}f)(t) = \mathrm{e}^{-t L} f_+$. The manipulations for the right hand side~$ \mathrm{e}^{-(\tau-t) L} f_-$ are similar, upon replacing~$t$ by~$\tau-t$ in the operators which appear. We consider the following ansatz, using a decomposition of the self-adjoint operator~$\mathcal{L}$ based on functional calculus, for functions~$\mathcal{F}_{0},\mathcal{F}_{1}:[0,\tau] \times [c_\phi,+\infty[ \to \mathbb{R}$ to be determined:
\[
Z_{0,+}(t) = \mathcal{F}_{0}(t,\mathcal{L})f_+, \qquad Z_{i,+}(t) = \partial_{x_i} \mathcal{F}_{1}(t,\mathcal{L}) f_+.
\]
Note that these functions satisfy the equation~\eqref{eq:Lions_A} with right hand side~$\mathrm{e}^{-t \mathcal{L}^{1/2}}f_+$ provided
\[
-\partial_t \mathcal{F}_{0}(t,\mathcal{L}) + \mathcal{L} \mathcal{F}_{1}(t,\mathcal{L}) = \mathrm{e}^{-t \mathcal{L}^{1/2}}.
\]
Moreover, in view of establishing the estimates~\eqref{eq:estimates_solution_divergence_equation} and in order to make use of the fact that~$P_{\mathcal{N},+}$ is bounded, it is convenient to rewrite~$\mathcal{F}_{0}(t,\mathcal{L})f_+$ and~$\mathcal{F}_1(t,\mathcal{L})f_+$ as the operators on~$\mathrm{L}^2(\mu)$
\begin{equation}
  \label{eq:from_F_to_mathcal_F}
  F_0(t,\mathcal{L}) = \mathcal{F}_0(t,\mathcal{L}) \mathrm{e}^{t L}, \qquad F_1(t,\mathcal{L}) = \mathcal{F}_1(t,\mathcal{L}) \mathrm{e}^{t L},
\end{equation}
applied to the function~$(P_{\mathcal{N},+} f)(t) = \mathrm{e}^{-t L}f_+$. This motivates looking for
\begin{equation}
  \label{eq:mathcal_F_as_polynomial}
  \mathcal{F}_0(t,\mathcal{L}) = P_{0}\left(\mathrm{e}^{-t L}\right), \qquad \mathcal{F}_1(t,\mathcal{L}) = P_1\left(\mathrm{e}^{-t L}\right),
\end{equation}
where~$P_0,P_1$ are polynomial functions satisfying the operator equation
\begin{equation}
  \label{eq:eq_P0_P1}
  -\frac{d}{dt} \left[ P_0\left(\mathrm{e}^{-t L}\right) \right] + \mathrm{L}^2 P_1\left(\mathrm{e}^{-t L}\right) = \mathrm{e}^{-t L},
\end{equation}
together with the boundary conditions
\begin{equation}
  \label{eq:BC_conditions_P0_P1}
  P_0(1) = P_0\left(\mathrm{e}^{-\tau L}\right) = P_1(1) = P_1\left(\mathrm{e}^{-\tau L}\right) = 0.
\end{equation}
In view of the boundary conditions~\eqref{eq:BC_conditions_P0_P1}, and the presence of the operators~$L$ and~$\mathrm{L}^2$ in~\eqref{eq:eq_P0_P1}, a natural ansatz is
\[
\left\{ \begin{aligned}
  P_0\left(\mathrm{e}^{-t L}\right) & = A_1 L^{-1} \left(1-\mathrm{e}^{-t L}\right)\left(\mathrm{e}^{-t L}-\mathrm{e}^{-\tau L}\right)\left(\mathrm{e}^{-t L} + A_2\right), \\
  P_1\left(\mathrm{e}^{-t L}\right) & = A_3 L^{-2} \left(1-\mathrm{e}^{-t L}\right)\left(\mathrm{e}^{-t L}-\mathrm{e}^{-\tau L}\right),
\end{aligned} \right.
\]
for operators~$A_1,A_2,A_3$ to be determined.

Note already that the operators~$P_0\left(\mathrm{e}^{-t L}\right)$ and~$P_1\left(\mathrm{e}^{-t L}\right)$ contain a composition with~$\mathrm{e}^{-t L}-\mathrm{e}^{-\tau L}$, which allows to further compose the operators with~$\mathrm{e}^{t L}$ in order to make sense of~\eqref{eq:from_F_to_mathcal_F}. In order to find the operators~$A_1,A_2,A_3$, which are assumed to be functions of~$L$ (and hence commute with all operators at hand), we evaluate~\eqref{eq:eq_P0_P1}: denoting by~$\theta_\star = \mathrm{e}^{-\tau L}$,
\[
A_1 \frac{d}{d\theta}\left[ (1-\theta)(\theta-\theta_\star)(\theta+A_2)\right] + A_3 (1-\theta)(\theta-\theta_\star) = \theta, 
\]
where the various functions are evaluated at $\theta = \mathrm{e}^{-t L}$. The left hand side of this equation is a second order polynomial in~$\theta$, so that the operators~$A_1,A_2,A_3$ are obtained by identifying the prefactors of the various terms~$\theta^a$ for~$a = 0,1,2$. More precisely,
\[
\left\{ \begin{aligned}
  -(A_1+A_3)\theta_\star + A_1 A_2 (1+\theta_\star) = 0, \\
  (2A_1+A_3)(1+\theta_\star) - 2A_1 A_2 = 1, \\
  -3A_1-A_3 = 0.
\end{aligned} \right.
\]
The last condition implies that~$A_3 = -3A_1$. Plugging this equality in the first condition leads to $A_2 = -2\mathrm{e}^{-\tau L}(1+\mathrm{e}^{-\tau L})^{-1}$. Finally, the second condition gives~$A_1(1+\theta_\star+2A_2) = -1$, so that
\[
A_1 = -\left[1+\mathrm{e}^{-\tau L} - 4 \mathrm{e}^{-\tau L}\left(1+\mathrm{e}^{-\tau L}\right)^{-1} \right]^{-1} = -\left(1+\mathrm{e}^{-\tau L}\right)\left(1-\mathrm{e}^{-\tau L}\right)^{-2}.
\]
Combining~\eqref{eq:from_F_to_mathcal_F} and~\eqref{eq:mathcal_F_as_polynomial} then leads to the expressions of~$F_0,F_1$ in~\eqref{eq:explicit_solution_divergence_equation}.

\subsection{Proof of the estimates~\eqref{eq:estimates_solution_divergence_equation}}
\label{sec:proof_estimates_divergence}

The estimate on the first term in~\eqref{eq:explicit_solution_divergence_equation} is a direct consequence of Lemma~\ref{lem:Lax_Milgram}.

\begin{proof}[Proof of Lemma~\ref{lem:Lax_Milgram}]
  The existence and uniqueness of the weak solution satisfying the estimate is established in~\cite[Lemma~2.4]{cao2019explicit}. 
  To make the constants precise, we revisit the steps of \cite{cao2019explicit}, just highlighting the differences. 
  \begin{enumerate}
    \item There exists a unique solution $u \in \mathrm{H}^1(\mathrm{U}_\tau \otimes \mu)$ to \eqref{eq:weak_formulation_LM}, which satisfies 
    \begin{eqnarray}\label{eq:ellipt2}
    \|u\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)} \leq C_{\mathrm P}^2 \,  \|f\|_{\mathrm{L}^{2}(\mathrm{U}_\tau \otimes \mu)}, \\
    \label{eq:ellipt3}
    \|\nabla_{t,x} u\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)} \leq C_{\mathrm P} \, \|f\|_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)}.
    \end{eqnarray}
    This is a consequence of Lax-Milgram's Theorem for the tensorised measure $\mathrm{U}_\tau \otimes \mu$ and Poincar\'e's inequality \eqref{poiphi}. 
    \item Thanks to \eqref{bochnerx} we have 
    $$\|\nabla^2_{t,x} u\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)} \leq 2 \|f\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)} + 2 c'_\phi \left(\sqrt{d} + 2 \max \left(8 c'_\phi , \sqrt{c''_\phi \,d} \right)  \,  \right)  \|\nabla_{t,x} u\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)}.$$
    \item Hence, by \eqref{eq:ellipt2},
    $$\| u\|^2_{\mathrm{H}^2(\mathrm{U}_\tau \otimes \mu)} \leq \left(2+ C^2_P + C^4_P + 2 \, C_{\mathrm{P}}^{-2} \, c'_\phi \left(\sqrt{d} + 2 \max \left(8 c'_\phi , \sqrt{c''_\phi \,d} \right)  \,  \right)\right) \, \|f\|^2_{\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu)}.$$
\end{enumerate}
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

  We conclude by showing that the sought boundary conditions are met, if $f \in \mathcal{N}^\perp$. 
  Consider an element~$w \in \mathcal{N}$, written as~\eqref{eq:decomposition_elements_N} for two smooth functions~$w_+,w_- \in \mathrm{L}^2_0(\mu)$ with compact support. Recall that~$(-\partial_t^2 + \mathcal{L})w = 0$. Then, the weak formulation~\eqref{eq:weak_formulation_LM} and integration by parts give
  \[
  \begin{aligned}
    \int_\mathcal{X} \left(u(\tau,\cdot)\partial_t w(\tau,\cdot) - u(0,\cdot)\partial_t w(0,\cdot)\right] d\mu & = \int_0^\tau \int_\mathcal{X} \left[ (\partial_t u)(\partial_t w) + \nabla_x u \cdot \nabla_x w \right]\, \mathrm{U}_\tau(dt) \, \mu(dx) \\
  & = \int_0^\tau \int_\mathcal{X} f w \, \mathrm{U}_\tau(dt) \, \mu(dx) = 0. 
  \end{aligned}
  \]
  the latter integral being equal to~0 since~$f \in \mathcal{N}^\perp$ by assumption. Therefore, 
  \[
  \int_\mathcal{X} \left[ u(\tau,\cdot) \mathcal{L}^{1/2}\left(-\mathrm{e}^{-\tau \mathcal{L}^{1/2}}w_+ + w_-\right) - u(0,\cdot) \mathcal{L}^{1/2}\left(-w_++\mathrm{e}^{-\tau \mathcal{L}^{1/2}}w_-\right)\right] d\mu = 0.
  \]
  Fixing~$w_-$ and choosing~$w_+ = \mathrm{e}^{-\tau \mathcal{L}^{1/2}}w_-$ leads to
  \[
  \int_\mathcal{X} u(\tau,\cdot) \mathcal{L}^{1/2}\left(1-\mathrm{e}^{-2\tau \mathcal{L}^{1/2}}\right)w_- \, d\mu = 0.
  \]
  Since this equality holds for any smooth~$w_- \in \mathrm{L}^2_0(\mu)$ with compact support, and the operator~$\mathcal{L}^{1/2}\left(1-\mathrm{e}^{-2\tau \mathcal{L}^{1/2}}\right)$ is invertible on~$\mathrm{L}^2_0(\mu)$, we can conclude that~$u(\tau,\cdot) = 0$. A similar reasoning allows to conclude that~$u(0) = 0$.
\end{proof}

We next show that the second and third terms in~\eqref{eq:explicit_solution_divergence_equation} satisfy~\eqref{eq:estimates_solution_divergence_equation}. Given that the two terms are extremely similar, we only treat the second one. Note first that the operators~$T_i = \partial_{x_i} L^{-1}$ are bounded on~$\mathrm{L}^2(\mu)$ and have an operator norm smaller than~1 since $T_1^* T_1+\dots+T_d^* T_d = 1$. Moreover, the operators~$\partial_{x_i} \partial_{x_j} L^{-2}$ for $1 \leq i,j \leq d$ are also bounded: 
$$\|\partial_{x_i} \partial_{x_j} L^{-2}\|_{\mathcal{B}(\mathrm{L}^2(\mathrm{U}_\tau \otimes \mu))} \leq \sqrt{ 2 + 2 \frac{c'_\phi}{C_{\mathrm P}^2} \left(\sqrt{d} + 2 \max \left(8 c'_\phi , \sqrt{c''_\phi \,d} \right)  \,  \right) },$$
by \eqref{bochnerx}; see also~\cite{dolbeault2015hypocoercivity}, as well as~\cite[Lemma~2.3]{cao2019explicit} and~\cite[Proposition~3.5]{bernard2020hypocoercivity}. The estimates~\eqref{eq:estimates_solution_divergence_equation} are therefore satisfied provided there exists~$C_\tau \in \mathbb{R}_+$ such that
\[
\begin{aligned}
  & \sup_{t \in [0,\tau]} \max \left( \left\| F_0(t,\mathcal{L}) \right\|_{\mathcal{B}(\mathrm{L}^2(\mu))}, \left\| \partial_t F_0(t,\mathcal{L}) \right\|_{\mathcal{B}(\mathrm{L}^2(\mu))}, \left\| L F_0(t,\mathcal{L}) \right\|_{\mathcal{B}(\mathrm{L}^2(\mu))} \right) \leq C_\tau, \\
  & \sup_{t \in [0,\tau]} \max \left( \left\| L F_1(t,\mathcal{L}) \right\|_{\mathcal{B}(\mathrm{L}^2(\mu))}, \left\| L \partial_t F_1(t,\mathcal{L}) \right\|_{\mathcal{B}(\mathrm{L}^2(\mu))}, \left\| \mathrm{L}^2 F_1(t,\mathcal{L}) \right\|_{\mathcal{B}(\mathrm{L}^2(\mu))} \right) \leq C_\tau.
\end{aligned}
\]
The explicit expressions of~$F_0,F_1$ show that~$C_\tau$ can be chosen of order
\[
\left\|1-\mathrm{e}^{-\tau L}\right\|_{\mathcal{B}(\mathrm{L}^2_0(\mu))}^{-2} = \left( \frac{1}{1-\mathrm{e}^{-\tau c_\phi}} \right)^2.
\]
Note that~$C_\tau$ scales as~$\tau^{-2}$ for~$\tau$ small.

