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Boundaries of random walks on graphs and groups with infinitely many ends. (English) Zbl 0723.60009
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.

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60G50 Sums of independent random variables; random walks
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