Fractal Properties of Random String Processes

Dongsheng Wu
Michigan State University

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

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

where $ \dot{W}(x,t)$ is a $ \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 talk, 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.

This talk is based on a joint work with Yimin Xiao.