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January 2002 Boundary and Entropy of Space Homogeneous Markov Chains
Vadim A. Kaimanovich, Wolfgang Woess
Ann. Probab. 30(1): 323-363 (January 2002). DOI: 10.1214/aop/1020107770

Abstract

We study the Poisson boundary ($\equiv$ representation of bounded harmonic functions) of Markov operators on discrete state spaces that are invariant under the action of a transitive group of permutations. This automorphism group is locally compact, but not necessarily discrete or unimodular. The main technical tool is the entropy theory which we develop along the same lines as in the case of random walks on countable groups, while, however, the implementation is different and exploits discreteness of the state space on the one hand and the path space of the induced random walk on the nondiscrete group on the other. Various new examples are given as applications, including a description of the Poisson boundary for random walks on vertex-transitive graphs with infinitely many ends and on the Diestel-Leader graphs.

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Vadim A. Kaimanovich. Wolfgang Woess. "Boundary and Entropy of Space Homogeneous Markov Chains." Ann. Probab. 30 (1) 323 - 363, January 2002. https://doi.org/10.1214/aop/1020107770

Information

Published: January 2002
First available in Project Euclid: 29 April 2002

zbMATH: 1021.60056
Digital Object Identifier: 10.1214/aop/1020107770

Subjects:
Primary: 60J50
Secondary: 05C25, 22F30, 60B15, 60G50

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.30 • No. 1 • January 2002
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