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

TL;DR

The paper develops a rigorous multi-scale framework for metastability in non-reversible finite-state Markov chains by establishing a $\\Gamma$-expansion of the measure-current large deviations rate functional. It demonstrates that the scaled rate functionals $\\theta^{(p)}_n I_n$ converge to coarse-grained rate functionals $I^{(p)}$ describing effective dynamics among metastable wells, starting with the initial scale $I^{(0)}$ and proceeding through a rooted-tree decomposition that encodes time-scale separation. It also analyzes when the Donsker-Varadhan or measure-current rate functionals uniquely determine the underlying dynamics, providing derivatives of the DV functional and linking them to the asymptotic variance of observables. Together with prior work, these results complete the hierarchical metastability program for finite-state Markov chains and offer a principled description of multi-scale metastable behavior. The framework is expected to facilitate accurate coarse-graining and analysis of complex metastable systems in statistical mechanics and related fields.

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 $I_n = \mathbf I^{(0)} \,+\, \sum_{1\le p\le \mathfrak q} (1/θ^{(p)}_n) \, \mathbf I^{(p)}$ for some rate functionals $\mathbf I^{(p)}$. The weights $θ^{(p)}_n$ 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 $Γ$-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.

Theorems & Definitions (37)

• Theorem 1.1
• Corollary 1.2
• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Remark 2.7
• proof : Proof of Proposition \ref{['p01']}