A sharp hypocoercive entropy decay estimate for underdamped Langevin dynamics

Abstract

We study the underdamped Langevin dynamics with invariant measure $\mu(\,\mathrm{d}x\,\mathrm{d}v)\propto \mathrm{e}^{-U(x)-\lvert v\rvert^2/2}\,\mathrm{d}x\,\mathrm{d}v$. Assume that the position marginal $\mu_x(\,\mathrm{d}x)\propto \mathrm{e}^{-U(x)}\,\mathrm{d}x$ satisfies a logarithmic Sobolev inequality with constant $\rho>0$, 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_\epsilon(g)=\operatorname{Ent}_\mu(g) +\epsilon\int \Pi_v(v\,g)\cdot\bigl(x-T_q(x)\bigr)\,\mu_x(\mathrm{d}x), \end{equation*} where $\Pi_v$ denotes averaging over the velocity variable against the standard Gaussian $\kappa(\mathrm{d}v)=(2\pi)^{-d/2}\mathrm{e}^{-\lvert v\rvert^2/2}\,\mathrm{d}v$, $q=\Pi_v g$ is the position marginal density of $g$, and $T_q$ is the Brenier optimal transport map from $q\mu_x$ to $\mu_x$. For friction $\gamma=\Gamma\sqrt\rho$ with $\Gamma>0$, 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\mid\mu) \leq \frac{1+\theta}{1-\theta}\,\mathrm{e}^{-\Lambda t}\,\operatorname{Ent}(p_0\mid\mu), \qquad t\ge0, \end{equation*} with rate \begin{equation*} \Lambda=\frac{\theta}{2(1+\theta)}\sqrt\rho, \qquad \theta=\min\Bigl\{\tfrac{\Gamma}{12},\tfrac{1}{4\Gamma}\Bigr\}. \end{equation*} In particular, the entropy convergence rate has optimal $\sqrt\rho$ order.