TY - JOUR
T1 - Discrete random walks on the group Sol
AU - Brieussel, Jérémie
AU - Tanaka, Ryokichi
N1 - Publisher Copyright:
© 2015, Hebrew University of Jerusalem.
PY - 2015/9/1
Y1 - 2015/9/1
N2 - 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.
AB - 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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U2 - 10.1007/s11856-015-1200-x
DO - 10.1007/s11856-015-1200-x
M3 - Article
AN - SCOPUS:84945338625
VL - 208
SP - 291
EP - 321
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
SN - 0021-2172
IS - 1
ER -