Multifractal properties of the sets of zeroes of Brownian paths

• Autorzy: Dmitry Dolgopyat , Vadim Sidorov
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 60J65: Brownian motion
• 60G17: Sample path properties

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1995
• Tom: 147
• Numer: 2
• Strony: 157-171

Daty

wydano

1995

otrzymano

1994-11-05

Twórcy

autor

Dmitry Dolgopyat

• Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08540, U.S.A. ,

autor

Vadim Sidorov

• Chair of Probability Theory, Moscow State University, Department of Mathematics and Mechanics, 119 899 Moscow, Russia

Bibliografia

• [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.