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 F. Hérau and developped by Dolbeault, Mouhot and Schmeiser, we show that the dynamics converges exponentially fast to equilibrium in the topologies $L^2(d\mu )$ and $L^2(W^{\ast }d\mu )$, where $\mu$ denotes the invariant probability measure and $W^{\ast }$ 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.

References in corpus (3)

• Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation
• From ballistic to diffusive behavior in periodic potentials
• On explicit $L^2$-convergence rate estimate for piecewise deterministic Markov processes in MCMC algorithms

Cited by in corpus (1)

• Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations