Mixing of metastable diffusion processes with Gibbs invariant distribution

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.