Metastability of reversible random walks in potential fields

TL;DR

This paper studies the metastable behavior of a class of reversible random walks in potential fields, focusing on how they transition among different wells of the potential as the discretization becomes finer.

Contribution

It provides a detailed analysis of the metastability phenomena for reversible random walks in potential fields, including the characterization of transition times and states.

Findings

Metastable transition times scale exponentially with N

Identification of dominant transition pathways between wells

Asymptotic behavior of the random walk in the potential landscape

Abstract

Let $\Xi$ be an open and bounded subset of $\bb R^d$, and let $F:\Xi\to\bb R$ be a twice continuously differentiable function. Denote by $\Xi_N$ th discretization of $\Xi$, $\Xi_N = \Xi \cap (N^{-1} \bb Z^d)$, and denote by $X_N(t)$ the continuous-time, nearest-neighbor, random walk on $\Xi_N$ which jumps from $\bs x$ to $\bs y$ at rate $ e^{-(1/2) N [F(\bs y) - F(\bs x)]}$. We examine in this article the metastable behavior of $X_N(t)$ among the wells of the potential $F$.