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Bounds on the $$L^ 2$$ spectrum for Markov chains and Markov processes: A generalization of Cheeger’s inequality. (English) Zbl 0716.60073
Summary: We prove a general version of J. Cheeger’s inequality [Probl. Analysis, Sympos. in Honor of Salomon Bochner, Princeton Univ. 1969, 195- 199 (1970; Zbl 0212.449)] 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.

##### MSC:
 60J05 Discrete-time Markov processes on general state spaces 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 15A42 Inequalities involving eigenvalues and eigenvectors 47B38 Linear operators on function spaces (general) 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60J25 Continuous-time Markov processes on general state spaces 60J27 Continuous-time Markov processes on discrete state spaces 82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
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