A sharp hypocoercive entropy decay estimate for underdamped Langevin dynamics

TL;DR

This paper establishes a sharp exponential decay estimate for the entropy in underdamped Langevin dynamics, utilizing a modified entropy approach with a Wasserstein entropy-current corrector, achieving an optimal rate proportional to the square root of the logarithmic Sobolev constant.

Contribution

The authors introduce a novel modified entropy method involving a Wasserstein entropy-current corrector to derive explicit, sharp entropy decay rates for underdamped Langevin dynamics.

Findings

Proved explicit entropy decay with rate depending on the logarithmic Sobolev constant.

Achieved entropy convergence rate with optimal order  ho.

Established decay estimates for initial laws with finite relative entropy.

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.