On the Lyapunov Foster Criterion and Poincaré Inequality for Reversible Markov Chains

This article 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. 1) It relies entirely on functional-analytic non-probabilistic arguments. 2) It makes explicit the connection between a Foster–Lyapunov function and Poincaré inequality. The proof is used to derive an explicit bound for the spectral gap. An extension to the nonreversible case is also presented.