Fractal properties of the random string processes

Details Fractal properties of the random string processes Fractal properties of the random string processes

2006-12-22

arxiv.org/abs/math/0612700v2

IMS Lecture Notes Monograph Series 2006, Vol. 51, 128-147

Mathematics

Probability

Published at http://dx.doi.org/10.1214/074921706000000806 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/p

Scientific paper

10.1214/074921706000000806

Let $\{u_t(x),t\ge 0, x\in {\mathbb{R}}\}$ be a random string taking values in ${\mathbb{R}}^d$, specified by the following stochastic partial differential equation [Funaki (1983)]: \[\frac{\partial u_t(x)}{\partial t}=\frac{{\partial}^2u_t(x)}{\partial x^2}+\dot{W},\] where $\dot{W}(x,t)$ is an ${\mathbb{R}}^d$-valued space-time white noise. Mueller and Tribe (2002) have proved necessary and sufficient conditions for the ${\mathbb{R}}^d$-valued process $\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}$ 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 $\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}$. We also consider the Hausdorff and packing dimensions of the range and graph of the string.

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