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Martin boundary of random walks with unbounded jumps in hyperbolic groups. (English) Zbl 1326.31006
Summary: Given a probability measure on a finitely generated group, its Martin boundary is a natural way to compactify the group using the Green function of the corresponding random walk. For finitely supported measures in hyperbolic groups, it is known since the work of Ancona and Gouëzel-Lalley that the Martin boundary coincides with the geometric boundary. The goal of this paper is to weaken the finite support assumption. We first show that, in any nonamenable group, there exist probability measures with exponential tails giving rise to pathological Martin boundaries. Then, for probability measures with superexponential tails in hyperbolic groups, we show that the Martin boundary coincides with the geometric boundary by extending Ancona’s inequalities. We also deduce asymptotics of transition probabilities for symmetric measures with superexponential tails.
Reviewer: Reviewer (Berlin)

31C35 Martin boundary theory
60J50 Boundary theory for Markov processes
60B99 Probability theory on algebraic and topological structures
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