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

In this note, we consider the underdamped Langevin dynamics with invariant measure $\mu (\mathrm{d}x\mathrm{d}v)\propto e^{-U(x)-\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}2}\mathrm{d}x\mathrm{d}v$. 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 ${\mathrm{\nabla }}^{2}U\ge -K\mathrm{I}\mathrm{d}$ for some $K\ge 0$. We revisit the modified $L^2$ method of Dolbeault–Mouhot–Schmeiser, employing a gap-shifted corrector \begin{equation*} A_m=(m- L_{$\mathrm{o}$})^{-1}(L_aΠ_v)^, \end{equation} where $L_{\mathrm{o}}={\mathrm{\Delta }}_x-\mathrm{\nabla }U\cdot {\mathrm{\nabla }}_x$ is the overdamped generator, $L_a$ is the generator of the Hamiltonian flow, and ${\mathrm{\Pi }}_v$ denotes averaging over the velocity variable. We establish an explicit hypocoercive $L^2$-convergence rate \begin{equation*} Λ=\frac{1}{6\Bigl(\sqrt{2+\frac{K}{2m}}+\sqrt{4+\frac{K}{2m}}\Bigr)}\sqrt{m}. \end{equation*} In particular, for convex $U$, this recovers the optimal $O(\sqrt{m})$ rate.