Algebraic sum of the image sets for a random string process

Abstract

Let $\{u_{t}(x)\colon t\geq 0{,}\ x\in \mathbb{R}\}$ be a random string taking values in $\mathbb{R}^{d}$. It is specified by the following stochastic partial differential equation, \begin{equation*} \frac{\partial u_{t}(x)}{\partial t} =\frac{\partial^{2}u_{t}(x)}{\partial x^{2}}+\dot{W}, \end{equation*} where $\dot{W}(x,t)$ is two-parameter white noise. The objective of the present paper is to study the fractal properties of the algebraic sum of the image sets for the random string process $\{u_{t}(x)\colon t\geq 0{,}\ x\in \mathbb{R}\}$. We obtain the Hausdorff and packing dimensions of the algebraic sum of the image sets of the string. We also consider the existence of the local times of the process $\{u_{s}(y)-u_{t}(x)\colon s,t\geq 0\colon x, y\in \mathbb{R}\}$, and find the Hausdorff and packing dimensions of the level sets for the process $\{u_{s}(y)-u_{t}(x)\colon s,t\geq 0; x, y\in \mathbb{R}\}$.