The Poisson boundary of covering Markov operators
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.
scientific article; zbMATH DE number 755539
scientific article