# 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

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topDolgopyat, 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

top- [1] K. Evertz, Laplacian fractals, Ph.D. thesis, Yale University, 1989.
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- [3] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, Wiley, 1970. Zbl0039.13201
- [4] I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen, 1971. Zbl0219.60027
- [5] K. Ito and H. McKean, Diffusion Processes and their Sample Paths, Springer, Berlin, 1965. Zbl0127.09503
- [6] G. M. Molchan, Multi-mono-fractal properties of Brownian zeroes, Proc. Russian Acad. Sci. 335 (1994), 424-427.
- [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] S. J. Taylor, The measure theory of random fractals, Math. Proc. Cambridge Philos. Soc. 100 (1986), 383-406. Zbl0622.60021
- [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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