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.
Publié le : 2002-01-14
Classification:
Markov chain,
random walk,
group,
intropy,
Poisson boundary,
60J50,
05C25,
22F30,
60B15,
60G50
@article{1020107770,
author = {Kaimanovich, Vadim A. and Woess, Wolfgang},
title = {Boundary and Entropy of Space Homogeneous Markov Chains},
journal = {Ann. Probab.},
volume = {30},
number = {1},
year = {2002},
pages = { 323-363},
language = {en},
url = {http://dml.mathdoc.fr/item/1020107770}
}
Kaimanovich, Vadim A.; Woess, Wolfgang. Boundary and Entropy of Space Homogeneous Markov Chains. Ann. Probab., Tome 30 (2002) no. 1, pp. 323-363. http://gdmltest.u-ga.fr/item/1020107770/