Mixing of metastable diffusion processes with Gibbs invariant distribution

TL;DR

This paper investigates the slow mixing behavior of metastable diffusion processes with Gibbs invariant distributions, providing precise asymptotics for mixing times using metastability theory.

Contribution

It offers a detailed analysis of mixing times for metastable diffusions with multiple equilibria, extending previous work by computing total variation distances and asymptotics.

Findings

Metastable transitions dominate mixing times.

Explicit asymptotic formulas for mixing times.

Total variation distance estimates for large time scales.

Abstract

In this article, we study the mixing properties of metastable diffusion processes which possess a Gibbs invariant distribution. For systems with multiple stable equilibria, so-called metastable transitions between these equilibria are required for mixing since the unique invariant distribution is concentrated on these equilibria. Consequently, these systems exhibit slower mixing compared to those with a unique stable equilibrium, as analyzed in Barrera and Jara (Ann. Appl. Probab. 30:1164--1208, 2020). Our proof is based on the theory of metastability, which is a primary tool for studying systems with multiple stable equilibria. Within this framework, we compute the total variation distance between the distribution of the diffusion process and its invariant distribution for any time scale larger than $\epsilon^{-1}$. Finally, we derive precise asymptotics for the mixing time.