A sharp hypocoercive entropy decay estimate for underdamped Langevin dynamics

Abstract

We study the underdamped Langevin dynamics with invariant measure $\hat{I}¼(\mathrm{d}x\mathrm{d}v)\propto {\mathrm{e}}^{-U(x)-|v{|}^2/2}\mathrm{d}x\mathrm{d}v$. Assume that the position marginal $\hat{I}{¼}_x(\mathrm{d}x)\propto {\mathrm{e}}^{-U(x)}\mathrm{d}x$ satisfies a logarithmic Sobolev inequality with constant $\ddot{I}>0$, and that $U$ is convex on ${\mathbb{R}}^d$ and satisfies some growth conditions. We introduce a modified entropy approach with a Wasserstein entropy-current corrector \begin{equation*} \mathcal H_ε(g)=\operatorname{Ent}_μ(g) +ε\int Î _v(v\,g)\cdot\bigl(x-T_q(x)\bigr)\,μ_x(\mathrm{d}x), \end{equation*} where ${\hat{I}}_v$ denotes averaging over the velocity variable against the standard Gaussian $\hat{I}º(\mathrm{d}v)=(2\ddot{I}€{)}^{-d/2}{\mathrm{e}}^{-|v{|}^2/2}\mathrm{d}v$, $q={\hat{I}}_{v}g$ is the position marginal density of $g$, and $T_q$ is the Brenier optimal transport map from $q\hat{I}{¼}_x$ to $\hat{I}{¼}_x$. For friction ${\hat{I}}^3=\hat{I}“\sqrt{\ddot{I}}$ with $\hat{I}“>0$, and for any initial law $p_0$ with finite relative entropy, if $p_t$ denotes the law of underdamped Langevin dynamics at time $t$, we establish the explicit entropy decay \begin{equation*} \operatorname{Ent}(p_t\midμ) \leq \frac{1+θ}{1-θ}\,\mathrm{e}^{-Λt}\,\operatorname{Ent}(p_0\midμ), \qquad t\ge0, \end{equation*} with rate \begin{equation*} Λ=\fracθ{2(1+θ)}\sqrtρ, \qquad θ=\min\Bigl\{\tfracΓ{12},\tfrac{1}{4Γ}\Bigr\}. \end{equation*} In particular, the entropy convergence rate has optimal $\sqrt{\ddot{I}}$ order.

25 pages; v2: typo correction and minor polishing