Bounds on the 𝐿² spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality

Abstract

We prove a general version of Cheeger’s inequality for discrete-time Markov chains and continuous-time Markovian jump processes, both reversible and nonreversible, with general state space. We also prove a version of Cheeger’s inequality for Markov chains and processes with killing. As an application, we prove L 2 {L^2} exponential convergence to equilibrium for random walk with inward drift on a class of countable rooted graphs.