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

We study the long-time convergence behavior of underdamped Langevin dynamics, when the spatial equilibrium satisfies a weighted Poincar\'e inequality, with a general velocity distribution, which allows for fat-tail or subexponential potential energies, and provide constructive and fully explicit estimates in $\mathrm{L}^2$-norm with $\mathrm{L}^\infty$ initial conditions. A key ingredient is a space-time weighted Poincar\'e--Lions inequality, which in turn implies a weak Poincar\'e--Lions inequality.