Weighted L ²-contractivity of Langevin dynamics with singular potentials

Convergence to equilibrium of underdamped Langevin dynamics is studied under general assumptions on the potential U allowing for singularities. By modifying the direct approach to convergence in L (2) pioneered by Herau and developed by Dolbeault et al, we show that the dynamics converges exponentially fast to equilibrium in the topologies L (2)(d mu) and L (2)(W* d mu), where mu denotes the invariant probability measure and W* is a suitable Lyapunov weight. In both norms, we make precise how the exponential convergence rate depends on the friction parameter gamma in Langevin dynamics, by providing a lower bound scaling as min(gamma, gamma (-1)). The results hold for usual polynomial-type potentials as well as potentials with singularities such as those arising from pairwise Lennard-Jones interactions between particles.