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Article dans une revue Israel Journal of Mathematics Année : 2015

Discrete random walks on the group SOL

Résumé

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 Erdös and Solomyak.
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Dates et versions

hal-01549918, version 1 (29-06-2017)

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Jeremie Brieussel, Ryokichi Tanaka. Discrete random walks on the group SOL. Israel Journal of Mathematics, 2015, 208 (1), pp.291-321. ⟨10.1007/s11856-015-1200-x⟩. ⟨hal-01549918⟩
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