Martin boundary of random walks with unbounded jumps in hyperbolic groups

Abstract

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 Gou{ë}zel-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.

Cited by in corpus (5)

• Deviation inequalities and CLT for random walks on acylindrically hyperbolic groups
• Random walks of infinite moment on free semigroups
• An embedding of the Morse boundary in the Martin boundary
• The Martin boundary of an extension by a hyperbolic group
• Positive harmonic functions of transformed random walks