Ergodic properties of quasi-Markovian generalized Langevin equations with configuration dependent noise and non-conservative force

TL;DR

This paper investigates the ergodic behavior of quasi-Markovian generalized Langevin equations with configuration-dependent noise and non-conservative forces, establishing conditions for unique invariant measures and exponential convergence.

Contribution

It introduces a novel ergodicity condition specifically for generalized Langevin equations with complex noise and force structures, advancing understanding of their long-term behavior.

Findings

Existence and uniqueness of smooth invariant distributions.

Exponential convergence in weighted $L^{ abla} spaces.

Validity of the central limit theorem for solutions.

Abstract

We discuss the ergodic properties of quasi-Markovian stochastic differential equations, providing general conditions that ensure existence and uniqueness of a smooth invariant distribution and exponential convergence of the evolution operator in suitably weighted $L^{\infty}$ spaces, which implies the validity of central limit theorem for the respective solution processes. The main new result is an ergodicity condition for the generalized Langevin equation with configuration-dependent noise and (non-)conservative force.