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 $\mu(\mathrm{d}x\,\mathrm{d}v) \propto e^{-U(x)-|v|^2/2}\,\mathrm{d}x\,\mathrm{d}v$. Assume that the position marginal $\mu_x(\mathrm{d}x)\propto e^{-U(x)}\,\mathrm{d}x$ satisfies a Poincar\'{e} 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 gap-shifted corrector \begin{equation*} A_m=(m- L_{\mathrm{o}})^{-1}(L_a\Pi_v)^*, \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 \begin{equation*} \Lambda=\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.