On the solution process for a stochastic fractional partial differential equation driven by space–time white noise

• Published in: Statistics & probability letters Vol. 81; no. 8; pp. 1161 - 1172
• Main Author: Wu, Dongsheng
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.08.2011; Elsevier
• Series: Statistics & Probability Letters
• Subjects: Exact sciences and technology; General topics; Hausdorff dimension; Hitting probability; Mathematics; Packing dimension; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Space–time white noise; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Statistics; Stochastic analysis; Stochastic fractional partial differential equation; Stochastic fractional partial differential equation Space-time white noise Hitting probability Hausdorff dimension Packing dimension; Hausdorff dimension; Stochastic fractional partial differential equation; Packing dimension; 28A80; Space–time white noise; Hitting probability; 60G17; 60H15; 60G15; Probability theory; Fractional process; Space time; Stochastic process; Cauchy problem; Statistical method; Space-time white noise; Point process; White noise; Differential operator
• ISSN: 0167-7152; 1879-2103
• DOI: 10.1016/j.spl.2011.03.012

Let { u ( t , x ) : t ≥ 0 , x ∈ R } be the solution process for the following Cauchy problem for the stochastic fractional partial differential equation taking values in R d : ∂ ∂ t u ( t , x ) = D α δ u ( t , x ) + W ̇ ( t , x ) , t > 0 , x ∈ R ; u ( 0 , x ) = u 0 ( x ) , where D α δ ( 1 < α < 3 , | δ | ≤ min { α − [ α ] , 2 + [ α ] 2 − α } ) is the fractional differential operator with respect to the spatial variable x (see below for a definition), W ̇ ( t , x ) is an R d -valued space–time white noise, and u 0 is an initial random datum defined on 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 Mueller and Tribe (2002) and Wu and Xiao (2006) for random string processes.