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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Martin boundaries of random walks: ends of trees and groups
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by Massimo A. Picardello and Wolfgang Woess PDF
Trans. Amer. Math. Soc. 302 (1987), 185-205 Request permission

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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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 302 (1987), 185-205
  • MSC: Primary 60J50; Secondary 05C05, 20F32, 60B15, 60J10
  • DOI: https://doi.org/10.1090/S0002-9947-1987-0887505-2
  • MathSciNet review: 887505