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September 2015 Martin boundary of random walks with unbounded jumps in hyperbolic groups
Sébastien Gouëzel
Ann. Probab. 43(5): 2374-2404 (September 2015). DOI: 10.1214/14-AOP938

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.

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Sébastien Gouëzel. "Martin boundary of random walks with unbounded jumps in hyperbolic groups." Ann. Probab. 43 (5) 2374 - 2404, September 2015. https://doi.org/10.1214/14-AOP938

Information

Received: 1 September 2013; Published: September 2015
First available in Project Euclid: 9 September 2015

zbMATH: 1326.31006
MathSciNet: MR3395464
Digital Object Identifier: 10.1214/14-AOP938

Subjects:
Primary: 31C35, 60B99, 60J50

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 5 • September 2015
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