TY - JOUR

T1 - Branching Random Walk with exponentially decreasing steps, and stochastically self-similar measures

AU - Benjamini, Itai

AU - Gurel-Gurevich, Ori

AU - Solomyak, Boris

PY - 2009/3

Y1 - 2009/3

N2 - We consider a Branching Random Walk on ℝ whose step size decreases by a fixed factor, 0 < λ < 1, with each turn. This process generates a random probability measure on ℝ; that is, the limit of uniform distribution among the 2n 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 λ > 1/2 the limit measure is almost surely (a.s.) absolutely continuous with respect to the Lebesgue measure, but for Pisot 1/λ it is a.s. singular; (2) for all λ > (√5-1)/2 the support of the measure is a.s. the closure of its interior; (3) for Pisot 1/λ the support of the measure is "fractured": it is a.s. disconnected, and the components of the complement are not isolated on both sides.

AB - We consider a Branching Random Walk on ℝ whose step size decreases by a fixed factor, 0 < λ < 1, with each turn. This process generates a random probability measure on ℝ; that is, the limit of uniform distribution among the 2n 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 λ > 1/2 the limit measure is almost surely (a.s.) absolutely continuous with respect to the Lebesgue measure, but for Pisot 1/λ it is a.s. singular; (2) for all λ > (√5-1)/2 the support of the measure is a.s. the closure of its interior; (3) for Pisot 1/λ the support of the measure is "fractured": it is a.s. disconnected, and the components of the complement are not isolated on both sides.

KW - Bernoulli convolutions

KW - Random fractal measures

UR - http://www.scopus.com/inward/record.url?scp=77950532938&partnerID=8YFLogxK

U2 - 10.1090/s0002-9947-08-04523-6

DO - 10.1090/s0002-9947-08-04523-6

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AN - SCOPUS:77950532938

SN - 0002-9947

VL - 361

SP - 1625

EP - 1643

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 3

ER -