Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60J50, 05C05, 20F32, 60B15, 60J10

Retrieve articles in all journals with MSC: 60J50, 05C05, 20F32, 60B15, 60J10

Additional Information

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

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia