Boundary and Entropy of Space Homogeneous Markov Chains
Vadim A. Kaimanovich and Wolfgang Woess
Source: Ann. Probab. Volume 30, Number 1
(2002), 323-363.
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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Permanent link to this document: http://projecteuclid.org/euclid.aop/1020107770
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Digital Object Identifier: doi:10.1214/aop/1020107770
Zentralblatt MATH identifier: 1021.60056
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IRMAR, UNIVERSITÉ DE RENNES-1 35042 RENNES FRANCE E-MAIL: kaimanov@univ-rennes1.fr INSTITUT FÜR MATHEMATIK TECHNISCHE UNIVERSITÄT GRAZ 8010 GRAZ AUSTRIA E-MAIL: woess@tugraz.at
