The Generalized Langevin Equation with a power-law memory in a nonlinear potential well

TL;DR

This paper investigates the long-term behavior of a generalized Langevin equation with power-law memory in nonlinear potential wells, revealing conditions for unique stationary distributions and invariant measures in different regimes.

Contribution

It extends previous results to power-law memory kernels, analyzing the existence and uniqueness of stationary distributions in both integrable and non-integrable cases.

Findings

Unique stationary distribution exists when the kernel is integrable.

Invariant probability measure exists even when the kernel is not integrable.

Method of asymptotic coupling does not apply in the non-integrable case.

Abstract

The generalized Langevin equation (GLE) is a stochastic integro-differential equation that has been used to describe the velocity of microparticles in viscoelastic fluids. In this work, we consider the large-time asymptotic properties of a Markovian approximation to the GLE in the presence of a wide class of external potential wells. The qualitative behavior of the GLE is largely determined by its memory kernel $K$, which summarizes the delayed response of the fluid medium on the particles past movement. When $K$ can be expressed as a finite sum of exponentials, it has been shown that long-term time-averaged properties of the position and velocity do not depend on $K$ at all. In certain applications, however, it is important to consider the GLE with a power law memory kernel. Using the fact that infinite sums of exponentials can have power law tails, we study the infinite-dimensional version of the Markovian GLE in a potential well. In the case where the memory kernel $K$ is integrable (i.e. in the asymptotically diffusive regime), we are able to extend previous results and show that there is a unique stationary distribution for the GLE system and that the long-term statistics of the position and velocity do not depend on $K$. However, when $K$ is not integrable (i.e. in the asymptotically subdiffusive regime), we are able to show the existence of an invariant probability measure but uniqueness remains an open question. In particular, the method of asymptotic coupling used in the integrable case to show uniqueness does not apply when $K$ fails to be integrable.