$\Gamma$-expansion of the measure-current large deviations rate functional of non-reversible finite-state Markov chains

TL;DR

This paper develops a $ ext{Gamma}$-expansion for the large deviations rate functional of non-reversible finite-state Markov chains, revealing the hierarchical metastable structure and dynamics at multiple time-scales.

Contribution

It introduces a novel $ ext{Gamma}$-expansion of the rate functional, characterizing metastability and hierarchical behavior in non-reversible Markov chains.

Findings

Decomposition of the rate functional into scale-dependent components.

Identification of conditions for the rate functional to determine the dynamics.

Calculation of derivatives of the rate functional extending previous results.

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 measure-current 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 $I_n = \mathbf I^{(0)} \,+\, \sum_{1\le p\le \mathfrak q} (1/\theta^{(p)}_n) \, \mathbf I^{(p)}$ for some rate functionals $\mathbf I^{(p)}$. The weights $\theta^{(p)}_n$ correspond to the time-scales at which the sequence of Markov chains $X^{(n)}_t$ evolves among the metastable wells, and the rate functionals $\mathbf I^{(p)}$ characterise the asymptotic Markovian dynamics among these wells. This expansion provides therefore an alternative description of the metastable behavior of a sequence of Markovian dynamics. Together with the results in \cite{bgl-24,l-gamma}, this work finishes the project of characterising the hierarchical metastable behavior of finite-state Markov chains by means of the $\Gamma$-expansion of large deviations rate functionals. In addition, we present optimal conditions under which the measure (Donsker-Varadhan) or the measure-current large deviations rate functional determines the original dynamics, and calculate the first and second derivatives of the measure large deviations rate functional, thereby generalising the results for i.i.d. random variables.