Branching random walk with exponentially decreasing steps, and stochastically self-similar measures
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- by Itai Benjamini, Ori Gurel-Gurevich and Boris Solomyak PDF
- Trans. Amer. Math. Soc. 361 (2009), 1625-1643 Request permission
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 âfracturedâ: it is a.s. disconnected, and the components of the complement are not isolated on both sides.References
- Matthias Arbeiter, Random recursive construction of self-similar fractal measures. The noncompact case, Probab. Theory Related Fields 88 (1991), no. 4, 497â520. MR 1105715, DOI 10.1007/BF01192554
- M. Arbeiter, Construction of random fractal measures by branching processes, Stochastics Stochastics Rep. 39 (1992), no. 4, 195â212. MR 1275122, DOI 10.1080/17442509208833775
- Krishna B. Athreya and Peter E. Ney, Branching processes, Die Grundlehren der mathematischen Wissenschaften, Band 196, Springer-Verlag, New York-Heidelberg, 1972. MR 0373040
- Itai Benjamini and Harry Kesten, Percolation of arbitrary words in $\{0,1\}^\textbf {N}$, Ann. Probab. 23 (1995), no. 3, 1024â1060. MR 1349161
- M.-J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, and J.-P. Schreiber, Pisot and Salem numbers, BirkhÀuser Verlag, Basel, 1992. With a preface by David W. Boyd. MR 1187044, DOI 10.1007/978-3-0348-8632-1
- Christian Bluhm, Fourier asymptotics of statistically self-similar measures, J. Fourier Anal. Appl. 5 (1999), no. 4, 355â362. MR 1700089, DOI 10.1007/BF01259376
- Jonathan M. Borwein and Roland Girgensohn, Functional equations and distribution functions, Results Math. 26 (1994), no. 3-4, 229â237. MR 1300602, DOI 10.1007/BF03323043
- Paul Erdös, On a family of symmetric Bernoulli convolutions, Amer. J. Math. 61 (1939), 974â976. MR 311, DOI 10.2307/2371641
- Paul Erdös, On the smoothness properties of a family of Bernoulli convolutions, Amer. J. Math. 62 (1940), 180â186. MR 858, DOI 10.2307/2371446
- PĂĄl Erdös, IstvĂĄn JoĂł, and Vilmos Komornik, Characterization of the unique expansions $1=\sum ^\infty _{i=1}q^{-n_i}$ and related problems, Bull. Soc. Math. France 118 (1990), no. 3, 377â390 (English, with French summary). MR 1078082
- Steven N. Evans, Polar and nonpolar sets for a tree indexed process, Ann. Probab. 20 (1992), no. 2, 579â590. MR 1159560
- K. J. Falconer, Random fractals, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 3, 559â582. MR 857731, DOI 10.1017/S0305004100066299
- Adriano M. Garsia, Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc. 102 (1962), 409â432. MR 137961, DOI 10.1090/S0002-9947-1962-0137961-5
- Paul Glendinning and Nikita Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett. 8 (2001), no. 4, 535â543. MR 1851269, DOI 10.4310/MRL.2001.v8.n4.a12
- Siegfried Graf, Statistically self-similar fractals, Probab. Theory Related Fields 74 (1987), no. 3, 357â392. MR 873885, DOI 10.1007/BF00699096
- John Hawkes, Trees generated by a simple branching process, J. London Math. Soc. (2) 24 (1981), no. 2, 373â384. MR 631950, DOI 10.1112/jlms/s2-24.2.373
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713â747. MR 625600, DOI 10.1512/iumj.1981.30.30055
- John E. Hutchinson and Ludger RĂŒschendorf, Random fractal measures via the contraction method, Indiana Univ. Math. J. 47 (1998), no. 2, 471â487. MR 1647916, DOI 10.1512/iumj.1998.47.1461
- Thomas Jordan, Mark Pollicott, and KĂĄroly Simon, Hausdorff dimension for randomly perturbed self affine attractors, Comm. Math. Phys. 270 (2007), no. 2, 519â544. MR 2276454, DOI 10.1007/s00220-006-0161-7
- Vilmos Komornik and Paola Loreti, Unique developments in non-integer bases, Amer. Math. Monthly 105 (1998), no. 7, 636â639. MR 1633077, DOI 10.2307/2589246
- Russell Lyons, Random walks and percolation on trees, Ann. Probab. 18 (1990), no. 3, 931â958. MR 1062053
- R. Daniel Mauldin and S. C. Williams, Random recursive constructions: asymptotic geometric and topological properties, Trans. Amer. Math. Soc. 295 (1986), no. 1, 325â346. MR 831202, DOI 10.1090/S0002-9947-1986-0831202-5
- Pedro Mendes and Fernando Oliveira, On the topological structure of the arithmetic sum of two Cantor sets, Nonlinearity 7 (1994), no. 2, 329â343. MR 1267692
- R. Olsen, Random Geometrically Graph Directed Self-similar Multifractals, Pittman Research Notes 307, Longman, 1994.
- N. Patzschke and U. ZĂ€hle, Self-similar random measures. IV. The recursive construction model of Falconer, Graf, and Mauldin and Williams, Math. Nachr. 149 (1990), 285â302. MR 1124811, DOI 10.1002/mana.19901490122
- Yuval Peres and Wilhelm Schlag, Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, Duke Math. J. 102 (2000), no. 2, 193â251. MR 1749437, DOI 10.1215/S0012-7094-00-10222-0
- Yuval Peres, Wilhelm Schlag, and Boris Solomyak, Sixty years of Bernoulli convolutions, Fractal geometry and stochastics, II (Greifswald/Koserow, 1998) Progr. Probab., vol. 46, BirkhĂ€user, Basel, 2000, pp. 39â65. MR 1785620
- Yuval Peres, KĂĄroly Simon, and Boris Solomyak, Absolute continuity for random iterated function systems with overlaps, J. London Math. Soc. (2) 74 (2006), no. 3, 739â756. MR 2286443, DOI 10.1112/S0024610706023258
- Yuval Peres and Boris Solomyak, Self-similar measures and intersections of Cantor sets, Trans. Amer. Math. Soc. 350 (1998), no. 10, 4065â4087. MR 1491873, DOI 10.1090/S0002-9947-98-02292-2
- Yuval Peres and Boris Solomyak, Problems on self-similar sets and self-affine sets: an update, Fractal geometry and stochastics, II (Greifswald/Koserow, 1998) Progr. Probab., vol. 46, BirkhĂ€user, Basel, 2000, pp. 95â106. MR 1785622
- A. RĂ©nyi, Probability theory, North-Holland Series in Applied Mathematics and Mechanics, Vol. 10, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1970. Translated by LĂĄszlĂł Vekerdi. MR 0315747
- Nikita Sidorov and Anatoly Vershik, Ergodic properties of the ErdĆs measure, the entropy of the golden shift, and related problems, Monatsh. Math. 126 (1998), no. 3, 215â261. MR 1651776, DOI 10.1007/BF01367764
- KĂĄroly Simon and Hajnal R. TĂłth, The absolute continuity of the distribution of random sums with digits $\{0,1,\dots ,m-1\}$, Real Anal. Exchange 30 (2004/05), no. 1, 397â409. MR 2127546
- Boris Solomyak, On the random series $\sum \pm \lambda ^n$ (an ErdĆs problem), Ann. of Math. (2) 142 (1995), no. 3, 611â625. MR 1356783, DOI 10.2307/2118556
- U. ZĂ€hle, Self-similar random measures. I. Notion, carrying Hausdorff dimension, and hyperbolic distribution, Probab. Theory Related Fields 80 (1988), no. 1, 79â100. MR 970472, DOI 10.1007/BF00348753
Additional Information
- Itai Benjamini
- Affiliation: Department of Theoretical Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel
- MR Author ID: 311800
- 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
- MR Author ID: 809980
- Boris Solomyak
- Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195
- MR Author ID: 209793
- Email: solomyak@math.washington.edu
- 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.
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 1625-1643
- MSC (2000): Primary 60J80; Secondary 60G57, 28A80
- DOI: https://doi.org/10.1090/S0002-9947-08-04523-6
- MathSciNet review: 2457411