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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Branching random walk with exponentially decreasing steps, and stochastically self-similar measures


Authors: Itai Benjamini, Ori Gurel-Gurevich and Boris Solomyak
Journal: Trans. Amer. Math. Soc. 361 (2009), 1625-1643
MSC (2000): Primary 60J80; Secondary 60G57, 28A80
Published electronically: October 23, 2008
MathSciNet review: 2457411
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Abstract: We consider a Branching Random Walk on $ \mathbb{R}$ whose step size decreases by a fixed factor, $ 0<\lambda<1$, with each turn. This process generates a random probability measure on $ \mathbb{R}$; that is, the limit of uniform distribution among the $ 2^n$ particles of the $ n$-th step. We present an initial investigation of the limit measure and its support. We show, in particular, that (1) for almost every $ \lambda>1/2$ the limit measure is almost surely (a.s.) absolutely continuous with respect to the Lebesgue measure, but for Pisot $ 1/\lambda$ it is a.s. singular; (2) for all $ \lambda> (\sqrt{5}-1)/2$ the support of the measure is a.s. the closure of its interior; (3) for Pisot $ 1/\lambda$ the support of the measure is ``fractured'': it is a.s. disconnected, and the components of the complement are not isolated on both sides.


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Additional Information

Itai Benjamini
Affiliation: Department of Theoretical Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel

Ori Gurel-Gurevich
Affiliation: Department of Theoretical Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel
Address at time of publication: Theory Group, Microsoft Research, One Microsoft Way, Redmond, Washington 98052

Boris Solomyak
Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195
Email: solomyak@math.washington.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04523-6
PII: S 0002-9947(08)04523-6
Keywords: Random fractal measures, Bernoulli convolutions
Received by editor(s): August 15, 2006
Received by editor(s) in revised form: April 6, 2007
Published electronically: October 23, 2008
Additional Notes: The research of the third author was partially supported by NSF grant DMS 0355187.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.