Metastability from the large deviations point of view: A $\Gamma$-expansion of the level two large deviations rate functional of non-reversible finite-state Markov chains

TL;DR

This paper provides a $ ext{Gamma}$-expansion of the level two large deviations rate functional for non-reversible finite-state Markov chains, revealing the metastable structure and multi-scale behavior of the chains.

Contribution

It introduces a novel $ ext{Gamma}$-expansion of the large deviations rate functional, linking metastability and time-scales in non-reversible Markov chains.

Findings

Decomposition of $I_n$ into sum of rate functionals with scale-dependent weights

Identification of metastable states via zero level sets of $I^{(p)}$

Analysis of metastable behavior through large deviations framework

Abstract

Consider a sequence of continuous-time Markov chains $(X^{(n)}_t:t\ge 0)$ evolving on a fixed finite state space $V$. Let $I_n$ be the level two large deviations rate functional for $X^{(n)}_t$, as $t\to\infty$. Under a hypothesis on the jump rates, we prove that $I_n$ can be written as $I_n = I^{(0)} \,+\, \sum_{1\le p\le q} (1/\theta^{(p)}_n) \, I^{(p)}$ for some rate functionals $I^{(p)}$. The weights $\theta^{(p)}_n$ correspond to the time-scales at which the sequence of Markov chains $X^{(n)}_t$ exhibit a metastable behavior, and the zero level sets of the rate functionals $I^{(p)}$ identify the metastable states.