Multifractal properties of the sets of zeroes of Brownian paths

Dmitry Dolgopyat; Vadim Sidorov

Fundamenta Mathematicae (1995)

  • Volume: 147, Issue: 2, page 157-171
  • ISSN: 0016-2736

Abstract

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We study Brownian zeroes in the neighborhood of which one can observe a non-typical growth rate of Brownian excursions. We interpret the multifractal curve for the Brownian zeroes calculated in [6] as the Hausdorff dimension of certain sets. This provides an example of the multifractal analysis of a statistically self-similar random fractal when both the spacing and the size of the corresponding nested sets are random.

How to cite

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Dolgopyat, Dmitry, and Sidorov, Vadim. "Multifractal properties of the sets of zeroes of Brownian paths." Fundamenta Mathematicae 147.2 (1995): 157-171. <http://eudml.org/doc/212080>.

@article{Dolgopyat1995,
abstract = {We study Brownian zeroes in the neighborhood of which one can observe a non-typical growth rate of Brownian excursions. We interpret the multifractal curve for the Brownian zeroes calculated in [6] as the Hausdorff dimension of certain sets. This provides an example of the multifractal analysis of a statistically self-similar random fractal when both the spacing and the size of the corresponding nested sets are random.},
author = {Dolgopyat, Dmitry, Sidorov, Vadim},
journal = {Fundamenta Mathematicae},
keywords = {independent random variables; Brownian motion; local time; Hausdorff dimension; self-similarity},
language = {eng},
number = {2},
pages = {157-171},
title = {Multifractal properties of the sets of zeroes of Brownian paths},
url = {http://eudml.org/doc/212080},
volume = {147},
year = {1995},
}

TY - JOUR
AU - Dolgopyat, Dmitry
AU - Sidorov, Vadim
TI - Multifractal properties of the sets of zeroes of Brownian paths
JO - Fundamenta Mathematicae
PY - 1995
VL - 147
IS - 2
SP - 157
EP - 171
AB - We study Brownian zeroes in the neighborhood of which one can observe a non-typical growth rate of Brownian excursions. We interpret the multifractal curve for the Brownian zeroes calculated in [6] as the Hausdorff dimension of certain sets. This provides an example of the multifractal analysis of a statistically self-similar random fractal when both the spacing and the size of the corresponding nested sets are random.
LA - eng
KW - independent random variables; Brownian motion; local time; Hausdorff dimension; self-similarity
UR - http://eudml.org/doc/212080
ER -

References

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  1. [1] K. Evertz, Laplacian fractals, Ph.D. thesis, Yale University, 1989. 
  2. [2] K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985. Zbl0587.28004
  3. [3] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, Wiley, 1970. Zbl0039.13201
  4. [4] I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen, 1971. Zbl0219.60027
  5. [5] K. Ito and H. McKean, Diffusion Processes and their Sample Paths, Springer, Berlin, 1965. Zbl0127.09503
  6. [6] G. M. Molchan, Multi-mono-fractal properties of Brownian zeroes, Proc. Russian Acad. Sci. 335 (1994), 424-427. 
  7. [7] S. J. Taylor, The α-dimensional measure on the graph and set of zeroes of a Brownian path, Proc. Cambridge Philos. Soc. 51 (1953), 31-39. 
  8. [8] S. J. Taylor, The measure theory of random fractals, Math. Proc. Cambridge Philos. Soc. 100 (1986), 383-406. Zbl0622.60021
  9. [9] S. J. Taylor and J. G. Wendel, The exact Hausdorff measure of the zero set of a stable process, Z. Wahrsch. Verw. Gebiete 6 (1966), 170-180. Zbl0178.52702

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