Explicit convergence rates of underdamped Langevin dynamics under weighted and weak Poincaré-Lions inequalities

Abstract

We study the long-time behavior of the underdamped Langevin dynamics, in the case of so-called weak confinemen}. Indeed, any L^\infty distribution (in position and velocity) relaxes to equilibrium over time, and we quantify the convergence rate. In our situation, the spatial equilibrium distribution does not satisfy a Poincaré inequality. Instead, we assume a weighted Poincaré inequality, which allows for fat-tail or sub-exponential potential energies. We provide constructive and fully explicit estimates in L^2-norm for L^\infty initial data. A key-ingredient is a new space-time weighted Poincaré-Lions inequality, entailing, in turn, a weak Poincaré-Lions inequality