Boundaries of random walks on graphs and groups with infinitely many ends (Q756836)

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.