Abstract
The multifractal structure of zerosZ of Brownian motion is considered. For different measures onZ we find typical characteristics: the τ-function and the multifractal spectrumf(α). A dimensional interpretation off(α) is also discussed.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
A. A. Borovkov,Probability Theory (Nauka, 1986) [in Russian].
G. Brown, G. Michon, and J. Peyriére, On the multifractal analysis of measures.J. Stat. Phys. 66:791–790 (1992).
R. Cowley, Multifractal decompositions of Moran fractals,Adv. Math. 92:196–236 (1992).
W. Feller,An Introduction to Probability Theory and Its Applications, Vol. 2 (Wiley, New York, 1966).
U. Frisch and G. Parisi, On the singularity structure of fully developed turbulence, Appendix to U. Frisch, Fully developed turbulence and intermittency, inTurbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics (North-Holland, Amsterdam, 1985), pp. 84–88.
P. Grassberger, Generalizations of the Hausdorff dimension of fractal measures.Phys. Lett. 107A:101–105 (1985).
T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. J. Shraiman, Fractal measures and their singularities: The characterization of strange sets,Phys. Rev. A 33:1141–1151 (1986).
J. Hawkes and A. Truman, Statistics of local time and excursions for the Ornstein-Uhlenbeck process,Stochastic Analysis 1991, M. Barlow and N. Bingham, eds. (1991), pp. 91–103.
R. Holley and E. C. Waymire, Multifractal dimensions and scaling exponents for strongly bounded random cascades,Ann. Appl. Prob. 2:819–845 (1992).
K. Ito and H. P. McKean, Jr.,Diffusion Processes and Their Sample Paths (Springer-Verlag, 1965).
B. B. Mandelbrot, Multifractal measures, especially for the geophysicist,PAGEOPH 131:5–42 (1989).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Molchan, G.M. Multifractal analysis of Brownian zero set. J Stat Phys 79, 701–730 (1995). https://doi.org/10.1007/BF02184877
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02184877