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Title: Weighted L 2 -contractivity of Langevin dynamics with singular potentials
Abstract 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 Hérau and developed by Dolbeault et al , we show that the dynamics converges exponentially fast to equilibrium in the topologies L 2 (d μ ) and L 2 ( W * d μ ), where μ 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 γ in Langevin dynamics, by providing a lower bound scaling as min( γ , γ −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.
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