Random walks and Kuramochi boundaries of infinite networks
type: Item
instance of: scholarly article
title: Random walks and Kuramochi boundaries of infinite networks
zbMATH Open document ID: 1287.60057
author: Atsushi Kasue
published in: Osaka Journal of Mathematics
publication date: 2013-04-30T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1364390418
review text: \textit{A. Ancona}, \textit{R. Lyons} and \textit{Y. Peres} [Ann. Probab. 27, No. 2, 970--989 (1999; Zbl 0945.60063)] proved a result on the Royden decomposition; namely, they proved that a function of finite Dirichlet energy on a transient network converges along the random walk almost surely and in \(L^2\). It is introduced an admissible weight \(r\) on the set of edges of a given graph \(G= (V,E)\), which is a positive function satisfying a boundary condition. This, in turn, provides a distance on \(V\), called the geodesic distance. Furthermore, it is introduced the Kuramochi compactification of a connected infinite network \((V,E,r)\). The present paper deals with the Kuramochi boundary of a transient network. The author proves that the random walk converges almost surely to a random variable valued in the harmonic boundary. Furthermore, it is shown that a function of finite Dirichlet energy converges along the random walk to a random variable, almost surely and in \(L^2\).
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 6159566
zbMATH Keywords: random walks
reviewed by: Sophia L. Kalpazidou
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6159566