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Discrete random walks on the group Sol

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Abstract

The harmonic measure ν on the boundary of the group Sol associated to a discrete random walk of law µ was described by Kaimanovich. We investigate when it is absolutely continuous or singular with respect to Lebesgue measure. By ratio entropy over speed, we show that any countable non-abelian subgroup admits a finite first moment non-degenerate μ with singular harmonic measure ν. On the other hand, we prove that some random walks with finitely supported step distribution admit a regular harmonic measure. Finally, we construct some exceptional random walks with arbitrarily small speed but singular harmonic measures. The two later results are obtained by comparison with Bernoulli convolutions, using results of Erdős and Solomyak.

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Authors and Affiliations

  1. Université Montpellier 2, Place E. Bataillon cc 051, 34095, Montpellier, France

    Jérémie Brieussel

  2. Tohoku University, Aoba-ku, Aramaki, Aoba 6-3, 980-8578, Sendai, Japan

    Ryokichi Tanaka

Authors
  1. Jérémie Brieussel
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  2. Ryokichi Tanaka
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Correspondence to Jérémie Brieussel.

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Brieussel, J., Tanaka, R. Discrete random walks on the group Sol. Isr. J. Math. 208, 291–321 (2015). https://doi.org/10.1007/s11856-015-1200-x

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  • DOI: https://doi.org/10.1007/s11856-015-1200-x

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