Isoperimetric inequalities and Markov chains. (English) Zbl 0573.60059

Consider the stochastic matrix A of a reversible Markov chain as acting on a sequence space. An inequality between certain norms on this space is shown to be (nearly) equivalent to a certain rate of decay for powers of A. The results are related to probabilities on finite groups, to graph structures on compact manifolds, and to Sobolev and isoperimetric inequalities on groups. The proofs use Green’s functions, and an integral bound for Dirichlet forms. A criterion is obtained, using the norms, for transience of the Markov chain.
Reviewer: B.D.Craven


60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
46A45 Sequence spaces (including Köthe sequence spaces)
Full Text: DOI


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