Dimensional properties of the harmonic measure for a random walk on a hyperbolic group
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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.References
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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
- Received by editor(s): December 16, 2004
- Received by editor(s) in revised form: June 17, 2005
- Published electronically: January 26, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2286061