On explicit $L^2$-convergence rate estimate for underdamped Langevin dynamics

Abstract

We provide a refined explicit estimate of exponential decay rate of underdamped Langevin dynamics in $L^2$ distance, based on a framework developed in [1]. To achieve this, we first prove a Poincaré-type inequality with Gibbs measure in space and Gaussian measure in momentum. Our estimate provides a more explicit and simpler expression of decay rate; moreover, when the potential is convex with Poincaré constant $m\ll 1$, our estimate shows the decay rate of $O(\sqrt{m})$ after optimizing the choice of friction coefficient, which is much faster than $m$ for the overdamped Langevi dynamics.

Minor changes, to appear in Archive for Rational Mechanics and Analysis

References in corpus (3)

• Geometric ergodicity of Langevin dynamics with Coulomb interactions
• Longtime convergence of the Temperature-Accelerated Molecular Dynamics Method
• Weighted $L^2$-contractivity of Langevin dynamics with singular potentials

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