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

We study the long-time behavior of the underdamped Langevin dynamics, in the case of so-called \emph{weak confinement}. Indeed, any ${\mathrm{L}}^{\mathrm{\infty }}$ 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 ${\mathrm{L}}^2$-norm for ${\mathrm{L}}^{\mathrm{\infty }}$ initial data. A key-ingredient is a new space-time weighted Poincaré–Lions inequality, entailing, in turn, a weak Poincaré–Lions inequality.