Sharp hypocoercive convergence estimates for underdamped Langevin dynamics via the modified $L^2$ method

Abstract

In this note, we consider the underdamped Langevin dynamics with invariant measure $\mu (\mathrm{d}x\mathrm{d}v)\propto e^{-U(x)-|v{|}^2/2}\mathrm{d}x\mathrm{d}v$. Assume that the position marginal ${\mu }_x(\mathrm{d}x)\propto e^{-U(x)}\mathrm{d}x$ satisfies a Poincaré inequality with constant $m>0$, and that ${\nabla }^{2}U\ge -K\mathrm{Id}$ for some $K\ge 0$. We revisit the modified $L^2$ method of Dolbeault--Mouhot--Schmeiser, employing a shifted corrector \begin{equation*} A_α=(α- L_{\mathrm o})^{-1}(L_aΠ_v)^*, \qquad α\ge0, \end{equation*} where $L_{\mathrm{o}}={\Delta }_x-\nabla U\cdot {\nabla }_x$ is the overdamped generator, $L_a$ is the generator of the Hamiltonian flow, and ${\Pi }_v$ denotes averaging over the velocity variable. We establish an explicit hypocoercive $L^2$-convergence rate ${\Lambda }_{\alpha ,\gamma }$ for every shift $\alpha \ge 0$ and friction coefficient $\gamma >0$, and show that, for each fixed $\gamma$, the rate is maximized at $\alpha =0$. Optimizing further over $\gamma$ gives \begin{equation*} Λ_{0,γ_*} = \frac{\sqrt m} {2\left(2+\sqrt{2+\frac{2K}{m}} +\sqrt{6+\frac{2K}{m}}\right)}\,, \qquad \text{at} \quad γ_*=\sqrt{6m+2K}. \end{equation*} For convex $U$, this recovers the optimal $O(\sqrt{m})$ rate.

revised with improved results; 14pages