Convergence rates for nonequilibrium Langevin dynamics

TL;DR

This paper investigates how quickly nonequilibrium Langevin dynamics converge to their stationary state, analyzing the effects of perturbations and limits, with theoretical and numerical validation of convergence rates.

Contribution

It introduces a perturbative hypocoercive approach to quantify convergence rates for nonequilibrium Langevin dynamics, including Hamiltonian and overdamped limits.

Findings

Exponential convergence rates are established for nonequilibrium Langevin dynamics.

Maximal perturbation magnitudes are quantified relative to friction.

Numerical results confirm theoretical bounds on spectral gaps.

Abstract

We study the exponential convergence to the stationary state for nonequilibrium Langevin dynamics, by a perturbative approach based on hypocoercive techniques developed for equilibrium Langevin dynamics. The Hamiltonian and overdamped limits (corresponding respectively to frictions going to zero or infinity) are carefully investigated. In particular, the maximal magnitude of admissible perturbations are quantified as a function of the friction. Numerical results based on a Galerkin discretization of the generator of the dynamics confirm the theoretical lower bounds on the spectral gap.