Non-equilibrium functional inequalities for finite Markov chains

TL;DR

This paper develops generalized functional inequalities for finite Markov chains applicable to non-equilibrium measures, providing tools for analyzing convergence and coarse-graining errors in complex, non-reversible systems.

Contribution

It introduces non-equilibrium versions of classical inequalities, proves their stability, and applies them to quantify errors and assess coarse-graining in non-reversible Markov chains.

Findings

Established continuity of inequality constants with respect to measures

Derived explicit lower bounds for the inequalities

Provided quantitative error estimates for coarse-graining processes

Abstract

Functional inequalities such as the Poincar\'e and log-Sobolev inequalities quantify convergence to equilibrium in continuous-time Markov chains by linking generator properties to variance and entropy decay. However, many applications, including multiscale and non-reversible dynamics, require analysing probability measures that are not at equilibrium, where the classical theory tied to steady states no longer applies. We introduce generalised versions of these inequalities for arbitrary positive measures on a finite state space, retaining key structural properties of their classical counterparts. In particular, we prove continuity of the associated constants with respect to the reference measure and establish explicit positive lower bounds. As an application, we derive quantitative coarse-graining error estimates for non-reversible Markov chains, both with and without explicit scale separation, and propose a quantitative criterion for assessing the quality of coarse-graining maps.