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On the solution process for a stochastic fractional partial differential equation driven by space-time white noise. (English) Zbl 1220.60038
Summary: Let $\{u(t,x): t\ge 0$, $x\in\Bbb R$ be the solution process for the following Cauchy problem for the stochastic fractional partial differential equation taking values in $\Bbb R^d$: $$\frac{\partial}{\partial t}u(t,x)={\frak D}_\alpha^\delta u(t,x)+\dot W(t,x), \quad t>0,\ x\in\Bbb R; \qquad u(0,x)= u^0(x),$$ where ${\frak D}_\alpha^\delta$ $(1<\alpha <3$, $|\delta|\le\min\{\alpha-[\alpha]$, $2+[\alpha]_2-\alpha\})$ is the fractional differential operator with respect to the spatial variable $x$, $\dot w(t,x)$ is an $\Bbb R^d$-valued space-time white noise, and $u^0$ is an initial random datum defined on $\Bbb R$. In this paper, we study the sample path properties of the solution process. We first find the dimensions in which the process hits points, and then determine the Hausdorff and packing dimensions of the range, the graph and the level sets of the process. Our results generalize those of {\it C. Mueller} and {\it Tribe} [Electron. J. Probab. 7, Paper No. 10 (2002; Zbl 1010.60059)] and {\it D. Wu} and {\it Y. Xiao} [in: Giné, Evarist (ed.) et al., High dimensional probability. Beachwood, OH: IMS, Institute of Mathematical Statistics. Institute of Mathematical Statistics Lecture Notes -- Monograph Series 51, 128--147 (2006; Zbl 1120.60040)] for random string processes.

MSC:
60H15Stochastic partial differential equations
60G22Fractional processes, including fractional Brownian motion
60G15Gaussian processes
60G17Sample path properties
28A80Fractals
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References:
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