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 (dµ) and L 2 (W * dµ), 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.

A functional Law of the Iterated Logarithm for weakly hypoelliptic diffusions at time zero

“…the drift is not needed, in order to reach all directions of the phase space. Weakly hypoelliptic diffusions arise in a number of natural settings, from finite-dimensional stochastic models in turbulence, see [5,6,7,14,16,21,34,13,39], to canonical models in statistical mechanics and machine learning, see [9,12,17,20,30,31], where local almost sure behavior at time zero is not understood precisely but is nevertheless important. Usually, one can estimate the behavior at time zero by finding the support of the process using control theory via the Support Theorems [37,38].…”