On the Lyapunov Foster criterion and Poincar\'e inequality for Reversible Markov Chains

TL;DR

This paper provides an elementary, functional-analytic proof of stochastic stability for reversible Markov chains based on Foster-Lyapunov conditions, linking them to Poincaré inequalities and spectral gap bounds, with extensions to non-reversible cases.

Contribution

It introduces a simple, non-probabilistic proof connecting Foster-Lyapunov functions to Poincaré inequalities and spectral gaps, including an extension to non-reversible Markov chains.

Findings

Explicit spectral gap bounds derived from Foster-Lyapunov functions

Connection established between Foster-Lyapunov functions and Poincaré inequalities

Extension of results to non-reversible Markov chains

Abstract

This paper presents an elementary proof of stochastic stability of a discrete-time reversible Markov chain starting from a Foster-Lyapunov drift condition. Besides its relative simplicity, there are two salient features of the proof: (i) it relies entirely on functional-analytic non-probabilistic arguments; and (ii) it makes explicit the connection between a Foster-Lyapunov function and Poincar\'e inequality. The proof is used to derive an explicit bound for the spectral gap. An extension to the non-reversible case is also presented.