Boundaries of random walks on graphs and groups with infinitely many ends
type: Item
instance of: scholarly article
title: Boundaries of random walks on graphs and groups with infinitely many ends
zbMATH Open document ID: 0723.60009
DOI: 10.1007/BF02764985
author: Wolfgang Woess
published in: Israel Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: Given an irreducible random walk \((Z_ n)\) on a locally finite graph G with infinitely many ends, whose transition function is invariant with respect to a closed subgroup \(\Gamma\) of automorphisms of G acting transitively on the vertex set of G, the author studies the asymptotic behavior of \((Z_ n)\) on the space \(\Omega\) of ends of G. Apart of a special case (i.e. if \(\Gamma\) is amenable) \(\Omega\) can be shown to be a Furstenberg boundary, \((Z_ n)\) converges almost surely (towards \(\Omega\)), and the corresponding Dirichlet problem can be solved. If \((Z_ n)\) has finite range, then \(\Omega\) can be identified with the Poisson boundary. Some of the results are applied to discrete groups with finitely many ends.
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 4192743
zbMATH Keywords: probability theory on graphs
zbMATH Keywords: potential theory of Markov processes
zbMATH Keywords: random walk
zbMATH Keywords: Dirichlet problem
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02764985
OpenAlex ID: W2073988734
description: scientific article; zbMATH DE number 4192743