Finite Markov chains coupled to general Markov processes and an application to metastability II

TL;DR

This paper analyzes the relationship between a diffusion process perturbed by small noise and an associated Markov chain, focusing on a one-dimensional double-well potential to understand metastability and eigenvalue connections.

Contribution

It provides a detailed analysis of the coupling between diffusion and Markov chains in the context of metastability for a double-well potential.

Findings

Established explicit relations between diffusion and Markov chain in metastable states

Analyzed eigenvalue correspondence for the coupled processes

Provided insights into the approximation of diffusion behavior by Markov chains

Abstract

We consider a diffusion given by a small noise perturbation of a dynamical system driven by a potential function with a finite number of local minima. The classical results of Freidlin and Wentzell show that the time this diffusion spends in the domain of attraction of one of these local minima is approximately exponentially distributed and hence the diffusion should behave approximately like a Markov chain on the local minima. By the work of Bovier and collaborators, the local minima can be associated with the small eigenvalues of the diffusion generator. In Part I of this work, by applying a Markov mapping theorem, we used the eigenfunctions of the generator to couple this diffusion to a Markov chain whose generator has eigenvalues equal to the eigenvalues of the diffusion generator that are associated with the local minima and established explicit formulas for conditional probabilities associated with this coupling. The fundamental question now becomes to relate the coupled Markov chain to the approximate Markov chain suggested by the results of Freidlin and Wentzel. In this paper, we take up this question and provide a complete analysis of this relationship in the special case of a double-well potential in one dimension.