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

Abstract

We provide a new explicit estimate of exponential decay rate of underdamped Langevin dynamics in $L^2$ distance. To achieve this, we first prove a Poincar'{e}-type inequality with Gibbs measure in space and Gaussian measure in momentum. Our new estimate provides a more explicit and simpler expression of decay rate; moreover, when the potential is convex with Poincar'{e} constant $m \ll 1$, our new estimate offers the decay rate of $\mathcal{O}(\sqrt{m})$ after optimizing the choice of friction coefficient, which is much faster compared to $\mathcal{O}(m)$ for the overdamped Langevin dynamics.