Explicit convergence rates of underdamped Langevin dynamics under weighted and weak Poincar\'e--Lions inequalities

We study the long-time behavior of the underdamped Langevin dynamics, in the case of so-called \emph{weak confinement}. Indeed, any $\mathrm{L}^\infty$ 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 Poincar\'e inequality. Instead, we assume a weighted Poincar\'e inequality, which allows for fat-tail or sub-exponential potential energies. We provide constructive and fully explicit estimates in $\mathrm{L}^2$-norm for $\mathrm{L}^\infty$ initial data. A key-ingredient is a new space-time weighted Poincar\'e--Lions inequality, entailing, in turn, a weak Poincar\'e--Lions inequality.