A class of SPDE driven by fractional white noise (Q2702397)

The following stochastic partial differential equation NEWLINE\[NEWLINE {\partial\over \partial t} u(t,x) ={1\over 2} \Delta u(t,x) +{\mathcal A}_t u(t,x) \diamond \omega^H(t,x) NEWLINE\]NEWLINE for \(t>0\) and \(x\in\mathbb R^d\) subject to smooth deterministic initial conditions is studied. Here \({\mathcal A}_t\) is a first order partial differential operator, \(\diamond\) denotes the Wick product, and \( \omega^H\) is some fractional white noise with Hurst multiparameter \((h_0,h_1,\ldots,h_d)\). This article generalizes older results for space-time (or standard) white noise [see \textit{J. Potthoff, \textit G. Våge} and \textit{H. Watanabe}, Appl. Math. Optimization 38, No. 1, 95-107 (1998; Zbl 0908.60056)] and the heat equation with multiplicative noise [see the author, ibid. 43, No.~2, 221-243 (2001)]. Denote \(|H|=h_1+\cdots+h_d\). The author proves for the the solution \(u\) that \(u(t,x)\in {\mathcal L}^2\) (i.e., \(E|u(t,x)|^2<\infty\)) provided \(4h_0+H>d+3\). Moreover, for any \(\kappa>(4h_0-2)/(4h_0+|H|-d-3)\) he obtains NEWLINE\[NEWLINE\limsup_{{t\to\infty}} ( t^{-\kappa}\log \sup\{E|u(t,x)|^2:\;x\in\mathbb R^d \}) <\infty.NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 0953.00049].