Metastable distributions of Markov chains with rare transitions

TL;DR

This paper analyzes Markov chains with small transition probabilities, describing their metastable distributions across different time scales as the transition rates vary with a small parameter.

Contribution

It provides a detailed description of metastable distributions for Markov chains with transition rates depending on a small parameter, under mild asymptotic assumptions.

Findings

Characterization of metastable distributions at various time scales

Dependence of distributions on the small parameter psilon

Conditions under which metastable behavior emerges

Abstract

In this paper we consider Markov chains with transition rates that depend on a small parameter $\varepsilon$. Under a mild assumption on the asymptotics of these transition rates, we describe the behavior of the chain at various $\varepsilon$-dependent time scales, i.e., we calculate the metastable distributions.