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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Martin boundaries of random walks: ends of trees and groups


Authors: Massimo A. Picardello and Wolfgang Woess
Journal: Trans. Amer. Math. Soc. 302 (1987), 185-205
MSC: Primary 60J50; Secondary 05C05, 20F32, 60B15, 60J10
MathSciNet review: 887505
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Abstract: Consider a transient random walk $ {X_n}$ on an infinite tree $ T$ whose nonzero transition probabilities are bounded below. Suppose that $ {X_n}$ is uniformly irreducible and has bounded step-length. (Alternatively, $ {X_n}$ can be regarded as a random walk on a graph whose metric is equivalent to the metric of $ T$.) The Martin boundary of $ {X_n}$ is shown to coincide with the space $ \Omega $ of all ends of $ T$ (or, equivalently, of the graph). This yields a boundary representation theorem on $ \Omega $ for all positive eigenfunctions of the transition operator, and a nontangential Fatou theorem which describes their boundary behavior. These results apply, in particular, to finitely supported random walks on groups whose Cayley graphs admit a uniformly spanning tree. A class of groups of this type is constructed.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0887505-2
PII: S 0002-9947(1987)0887505-2
Keywords: Random walk on a tree, Martin boundary, harmonic functions, ends of a tree, ends of a group
Article copyright: © Copyright 1987 American Mathematical Society