The Annals of Probability

Martin boundary of random walks with unbounded jumps in hyperbolic groups

Sébastien Gouëzel

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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 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.

Article information

Source
Ann. Probab., Volume 43, Number 5 (2015), 2374-2404.

Dates
Received: September 2013
First available in Project Euclid: 9 September 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1441792288

Digital Object Identifier
doi:10.1214/14-AOP938

Mathematical Reviews number (MathSciNet)
MR3395464

Zentralblatt MATH identifier
1326.31006

Subjects
Primary: 31C35: Martin boundary theory [See also 60J50] 60J50: Boundary theory 60B99: None of the above, but in this section

Keywords
Random walk hyperbolic group Martin boundary Gromov boundary infinite range local limit theorem

Citation

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


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References

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