Full $\Gamma$-expansion of reversible Markov chains level two large deviations rate functionals

TL;DR

This paper develops a comprehensive method to expand the large deviations rate functional for reversible Markov chains, capturing metastable behaviors across multiple time scales with explicit rate functionals.

Contribution

It introduces a full expansion technique for the level two large deviations rate functional, revealing metastable states and their associated time scales in reversible Markov chains.

Findings

Derived a full expansion of the rate functional $I_n$ in terms of scale-dependent components.

Identified metastable states as zero-level sets of the rate functionals.

Applied the method to random walks in potential fields to demonstrate its effectiveness.

Abstract

Let $\Xi_n \subset \mathbb R^d$, $n\ge 1$, be a sequence of finite sets and consider a $\Xi_n$-valued, irreducible, reversible, continuous-time Markov chain $(X^{(n)}_t:t\ge 0)$. Denote by $\mathscr P(\mathbb R^d) $ the set of probability measures on $\mathbb R^d$ and by $I_n\colon \mathscr P(\mathbb R^d) \to [0,+\infty)$ the level two large deviations rate functional for $X^{(n)}_t$ as $t\to\infty$. We present a general method, based on tools used to prove the metastable behaviour of Markov chains, to derive a full expansion of $I_n$ expressing it as $I_n = I^{(0)} \,+\, \sum_{1\le p\le q} (1/\theta^{(p)}_n)\, I^{(p)}$, where $I^{(p)}\colon \mathscr P(\mathbb R^d) \to [0,+\infty]$ represent rate functionals independent of $n$ and $\theta^{(p)}_n$ sequences such that $\theta^{(1)}_n \to\infty$, $\theta^{(p)}_n / \theta^{(p+1)}_n \to 0$ for $1\le p< q$. The speed $\theta^{(p)}_n$ corresponds to the time-scale at which the Markov chains $X^{(n)}_t$ exhibits a metastable behavior, and the $I^{(p-1)}$ zero-level sets to the metastable states. To illustrate the theory we apply the method to random walks in potential fields.