Martin boundary of random walks with unbounded jumps in hyperbolic groups

Sébastien Gouëzel (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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 non-amenable 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.

Mots clés: hyperbolic group; Random walk; Martin boundary; Gromov boundary; infinite range; local limit theorem

Domaines: Probabilités [math.PR]; Systèmes dynamiques [math.DS]

Sébastien Gouëzel. Martin boundary of random walks with unbounded jumps in hyperbolic groups. Annals of Probability, 2015, 43 (5), pp.2374-2404. ⟨10.1214/14-AOP938⟩. ⟨hal-00793093v2⟩