Asymptotic analysis for the generalized langevin equation

TL;DR

This paper provides a comprehensive asymptotic analysis of the generalized Langevin equation, exploring ergodicity, homogenization, and noise limits using hypoelliptic operator techniques and hypocoercivity theory.

Contribution

It introduces new results on the asymptotic behavior of quasi-Markovian GLEs, including ergodicity and invariance principles, with novel analytical methods.

Findings

Proves geometric ergodicity of the GLE solutions.

Establishes a homogenization theorem and invariance principle.

Analyzes short time asymptotics and white noise limits.

Abstract

Various qualitative properties of solutions to the generalized Langevin equation (GLE) in a periodic or a confining potential are studied in this paper. We consider a class of quasi-Markovian GLEs, similar to the model that was introduced in \cite{EPR99}. Geometric ergodicity, a homogenization theorem (invariance principle), short time asymptotics and the white noise limit are studied. Our proofs are based on a careful analysis of a hypoelliptic operator which is the generator of an auxiliary Markov process. Systematic use of the recently developed theory of hypocoercivity \cite{Vil04HPI} is made.