Download PDF - Multifractal properties of the sets of zeroes of Brownian pathsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Multifractal properties of the sets of zeroes of Brownian paths

Authors 1, 2

Affiliations

  1. Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08540, U.S.A.
  2. Chair of Probability Theory, Moscow State University, Department of Mathematics and Mechanics, 119 899 Moscow, Russia

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.

Keywords

ENG
independent random variables, Brownian motion, local time, Hausdorff dimension, self-similarity

Bibliography

  1. K. Evertz, Laplacian fractals, Ph.D. thesis, Yale University, 1989.
  2. K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985.
  3. W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, Wiley, 1970.
  4. I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen, 1971.
  5. K. Ito and H. McKean, Diffusion Processes and their Sample Paths, Springer, Berlin, 1965.
  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.
  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.

Additional information

http://matwbn.icm.edu.pl/ksiazki/fm/fm147/fm14724.pdf

Fundamenta Mathematicae
1995/147/2
Export to PBN, Opens in new tab
Pages:
157-171
Main language of publication
English
Received
1994-11-05
Published
1995
Exact and natural sciences