The Poisson boundary of covering Markov operators
type: Item
instance of: scholarly article
title: The Poisson boundary of covering Markov operators
zbMATH Open document ID: 0843.43001
DOI: 10.1007/BF02808195
published in: Israel Journal of Mathematics
publication date: 1996-08-05T00:00:00Z
review text: The aim of the paper under review is a systematic investigation on Poisson boundaries of Markov operators connected with groups acting on the state space. The Poisson boundary of a Markov operator \(P\) acting on \(L^\infty (X,m)\) is a measure space \((\Gamma, \nu)\) such that the space of bounded \(P\)-harmonic functions is isomorphic to \( L^\infty (\Gamma, \nu)\). Markov operators arising in the context of random walks on (locally compact) groups and homogeneous spaces and their Poisson boundaries have been extensively studied in the past. More recent investigations consider Brownian motions on covering Riemannian manifolds. In both cases the Poisson boundary is endowed with a natural action of a group. This leads to the definition of a ``covering Markov operator'' acting on a ``covering measure space \((\widetilde X, \widetilde m)\)'' which is endowed with the action of a (countable) deck group \(G\), and the corresponding quotient Markov operators acting on \((\widetilde X, \widetilde m)/G\). This definition includes previous examples. The aim of the paper is -- now more precisely -- a systematic investigation of Poisson boundaries of covering Markov operators. The above mentioned motivating examples are treated again in \S 5 pointing out especially the interrelations between Poisson boundaries of random walks on countable groups and of diffusions on covering Riemannian manifolds on the one hand and of the corresponding covering Markov operators on the other hand.
Mathematics Subject Classification ID: 43A10
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60J50
zbMATH DE Number: 755539
zbMATH Keywords: \(P\)-harmonic functions
zbMATH Keywords: locally compact groups
zbMATH Keywords: covering Markov operator
zbMATH Keywords: covering measure space
zbMATH Keywords: Poisson boundaries
zbMATH Keywords: Markov operators
zbMATH Keywords: measure space
zbMATH Keywords: random walks
zbMATH Keywords: homogeneous spaces
zbMATH Keywords: Brownian motions
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: deck group
author: Vadim A. Kaimanovich
reviewed by: Wilfried Hazod
MaRDI profile type: Publication
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The Poisson boundary of covering Markov operators (full work available at URL)
full work available at URL: https://doi.org/10.1007/bf02808195

The Poisson boundary of covering Markov operators (OpenAlex ID)
OpenAlex ID: W2064047457