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Dimensional properties of the harmonic measure for a random walk on a hyperbolic group


Author: Vincent Le Prince
Journal: Trans. Amer. Math. Soc. 359 (2007), 2881-2898
MSC (2000): Primary 60G50, 20F67, 28D20, 28A78
DOI: https://doi.org/10.1090/S0002-9947-07-04108-6
Published electronically: January 26, 2007
MathSciNet review: 2286061
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Abstract: This paper deals with random walks on isometry groups of Gromov hyperbolic spaces, and more precisely with the dimension of the harmonic measure $ \nu$ associated with such a random walk. We first establish a link of the form $ \dim \nu\leq h/l$ between the dimension of the harmonic measure, the asymptotic entropy $ h$ of the random walk and its rate of escape $ l$. Then we use this inequality to show that the dimension of this measure can be made arbitrarily small and deduce a result on the type of the harmonic measure.


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Additional Information

Vincent Le Prince
Affiliation: IRMAR, Université de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex, France
Email: vincent.leprince@univ-rennes1.fr

DOI: https://doi.org/10.1090/S0002-9947-07-04108-6
Keywords: Ergodic theory, random walk, hyperbolic group, harmonic measure, entropy
Received by editor(s): December 16, 2004
Received by editor(s) in revised form: June 17, 2005
Published electronically: January 26, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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