On Explicit L2 -Convergence Rate Estimate for Underdamped Langevin Dynamics

We provide a refined explicit estimate of the exponential decay rate of underdamped Langevin dynamics in the L² distance, based on a framework developed in Albritton et al. (Variational methods for the kinetic Fokker–Planck equation, arXiv arXiv:1902.04037 , 2019). To achieve this, we first prove a Poincaré-type inequality with a Gibbs measure in space and a Gaussian measure in momentum. Our estimate provides a more explicit and simpler expression of the decay rate; moreover, when the potential is convex with a Poincaré constant m≪ 1 , our estimate shows the decay rate of O(m) after optimizing the choice of the friction coefficient, which is much faster than m for the overdamped Langevin dynamics.