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

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.