A sharp hypocoercive entropy decay estimate for underdamped Langevin dynamics

Abstract: We study the underdamped Langevin dynamics with invariant measure $μ(\,\mathrm{d}x\,\mathrm{d}v)\propto \mathrm{e}^{-U(x)-\lvert v\rvert^2/2}\,\mathrm{d}x\,\mathrm{d}v$. Assume that the position marginal $μx(\,\mathrm{d}x)\propto \mathrm{e}^{-U(x)}\,\mathrm{d}x$ satisfies a logarithmic Sobolev inequality with constant $ρ>0$, 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 Hε(g)=\operatorname{Ent}_μ(g) +ε\int Π_v(v\,g)\cdot\bigl(x-T_q(x)\bigr)\,μ_x(\mathrm{d}x), \end{equation*} where $Π_v$ denotes averaging over the velocity variable against the standard Gaussian $κ(\mathrm{d}v)=(2π)^{-d/2}\mathrm{e}^{-\lvert v\rvert^2/2}\,\mathrm{d}v$, $q=Π_v g$ is the position marginal density of $g$, and $T_q$ is the Brenier optimal transport map from $qμ_x$ to $μ_x$. For friction $γ=Γ\sqrtρ$ with $Γ>0$, 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 $\sqrtρ$ order.