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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 exponential convergence to equilibrium for random walk with inward drift on a class of countable rooted graphs

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