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

Vincent Le Prince
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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 $ν$ associated with such a random walk. We first establish a link of the form $\dim ν\leq h/l$ 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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Vincent Le Prince. 2004-11-15. Dimensional properties of the harmonic measure for a random walk on a hyperbolic group. https://arxiv.org/abs/math/0411332

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