Discrete random walks on the group Sol

Jérémie Brieussel, Ryokichi Tanaka

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)291-321
Number of pages31
JournalIsrael Journal of Mathematics
Volume208
Issue number1
DOIs
Publication statusPublished - 2015 Sep 1

ASJC Scopus subject areas

  • Mathematics(all)

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