The Annals of Probability
- Ann. Probab.
- Volume 43, Number 5 (2015), 2374-2404.
Martin boundary of random walks with unbounded jumps in hyperbolic groups
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.
Ann. Probab., Volume 43, Number 5 (2015), 2374-2404.
Received: September 2013
First available in Project Euclid: 9 September 2015
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Gouëzel, Sébastien. Martin boundary of random walks with unbounded jumps in hyperbolic groups. Ann. Probab. 43 (2015), no. 5, 2374--2404. doi:10.1214/14-AOP938. https://projecteuclid.org/euclid.aop/1441792288