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 direct approach convergence in $L^2$ pioneered by F. H\'erau and developped Dolbeault, Mouhot Schmeiser, we show that converges exponentially fast topologies $L^2(d\mu)$ $L^2(W^* d\mu)$, where $\mu$ denotes invariant probability measure $W^*$ a suitable Lyapunov weight. In both norms, make precise how exponential rate depends friction parameter $\gamma$ dynamics, providing lower bound scaling as $\min(\gamma, \gamma^{-1})$. The results hold usual polynomial-type potentials well with singularities such those arising from pairwise Lennard-Jones interactions between particles.