随机弦过程的分形性质
Fractal properties of the random string processes
摘要：
令$\{u_t(x),t\ge 0, x\in {\mathbb{R}}\}$是一个随机字符串，取值于${\mathbb{R}}^d$中，由下面的随机偏微分方程[Funaki（1983）]指定：\[\frac{\partial u_t(x)}{\partial t}=\frac{{\partial}^2u_t(x)}{\partial x^2}+\dot{W},\]其中$\dot{W}(x,t)$是一个${\mathbb{R}}^d$值的时空白噪声。Mueller和Tribe（2002）证明了${\mathbb{R}}^d$值过程$\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}$有命中点和有二重点的充要条件.在本文中，我们继续他们的研究，通过确定的水平集的Hausdorff维数和包装的随机字符串过程$\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}$倍的集合。我们还考虑了Hausdorff维数和填充维数的范围和图的字符串。
英文摘要：
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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