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VOL. 51 | 2006 Fractal properties of the random string processes
Dongsheng Wu, Yimin Xiao

Editor(s) Evarist Giné, Vladimir Koltchinskii, Wenbo Li, Joel Zinn

Abstract

Let $\{u_t(x), t \ge 0,\, x \in \R \}$ be a random string taking values in $\R^d$, specified by the following stochastic partial differential equation (Funaki, 1983):

\frac{\partial u_t(x)}{\partial t}=\frac{{\partial}^2u_t(x)}{\partial x^2}+\dot{W},

here $\dot{W}(x,t)$ is an $\R^d$-valued space-time white noise.

Mueller and Tribe (2002) have proved necessary and sufficient conditions for the $\R^d$-valued process $\{u_t(x):t \ge 0,\,x \in \R\}$ to hit points and to have double points. In this paper, we continue their research by determining the Hausdorff and packing dimensions of the level sets and the sets of double times of the random string process $\{u_t(x):t \ge 0,\,x \in \R\}$. We also consider the Hausdorff and packing dimensions of the range and graph of the string.

Information

Published: 1 January 2006
First available in Project Euclid: 28 November 2007

zbMATH: 1120.60040
MathSciNet: MR2387765

Digital Object Identifier: 10.1214/074921706000000806

Subjects:
Primary: 60G15 , 60G17 , 60H15
Secondary: 28A80

Keywords: double times , graph , Hausdorff dimension , Level set , Packing dimension , random string process , ‎range‎ , stationary pinned string

Rights: Copyright © 2006, Institute of Mathematical Statistics

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Vol. 51 • 1 January 2006