Under LSI and convexity assumptions on the target, the relative entropy of underdamped Langevin dynamics decays at an explicit rate proportional to sqrt(ρ) with constant depending on a scaled friction parameter.
Sharp hypocoercive convergence estimates for underdamped Langevin dynamics via the modified $L^2$ method
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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 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.
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math.AP 3years
2026 3verdicts
UNVERDICTED 3representative citing papers
Proves space-time LSI and hypocoercive hypercontractivity for underdamped Langevin dynamics, giving Rényi divergence decay at rate O(sqrt(rho)) under tuned friction.
Mean-field underdamped Langevin dynamics achieves Nesterov acceleration for Wasserstein minimization of displacement-convex free energies by extending a linear-case result to the nonlinear setting.
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A sharp hypocoercive entropy decay estimate for underdamped Langevin dynamics
Under LSI and convexity assumptions on the target, the relative entropy of underdamped Langevin dynamics decays at an explicit rate proportional to sqrt(ρ) with constant depending on a scaled friction parameter.
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Space-Time Log-Sobolev Inequality and Hypocoercive Hypercontractivity for Underdamped Langevin Dynamics
Proves space-time LSI and hypocoercive hypercontractivity for underdamped Langevin dynamics, giving Rényi divergence decay at rate O(sqrt(rho)) under tuned friction.
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Nesterov acceleration for the Wasserstein minimization of displacement-convex free energies
Mean-field underdamped Langevin dynamics achieves Nesterov acceleration for Wasserstein minimization of displacement-convex free energies by extending a linear-case result to the nonlinear setting.