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 shifted corrector \begin{equation*} A_\alpha=(\alpha- L_{\mathrm o})^{-1}(L_a\Pi_v)^*, \qquad \alpha\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 $\alpha\ge0$ 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*} \Lambda_{0,\gamma_*} = \frac{\sqrt m} {2\left(2+\sqrt{2+\frac{2K}{m}} +\sqrt{6+\frac{2K}{m}}\right)}\,, \qquad \text{at} \quad \gamma_*=\sqrt{6m+2K}. \end{equation*} For convex $U$, this recovers the optimal $O(\sqrt m)$ rate.