zbMATH — the first resource for mathematics

Stochastic PDE’s with function-valued solutions. (English) Zbl 0990.60065
Clément, Ph. (ed.) et al., Infinite dimensional stochastic analysis. Proceedings of the colloquium, Amsterdam, Netherlands, February 11-12, 1999. Amsterdam: Royal Netherlands Academy of Arts and Sciences. Verh. Afd. Natuurkd., 1. Reeks, K. Ned. Akad. Wet. 52, 197-216 (2000).
This paper is concerned to the stochastic heat and wave equations \[ \begin{cases} {\partial u\over\partial t} (t,\theta)=\Delta u(t,\theta)+ {\partial W\over\partial t} (t,\theta), &t> 0,\;\theta\in \mathbb{R}^d,\\ u(0,\theta)= 0,\quad\theta\in \mathbb{R}^d,\end{cases} \] and \[ \begin{cases}{\partial^2u\over\partial t^2} (t,\theta)=\Delta u(t,\theta)+ {\partial W_\Gamma\over\partial t} (t,\theta), &t> 0,\;\theta\in \mathbb{R}^d,\\ u(0,\theta)= 0,\quad{\partial u\over\partial t} (0,\theta)= 0,\quad \theta\in \mathbb{R}^d,\end{cases} \] where \(W_\Gamma\) is a spatially homogeneous Wiener process with space correlation \(\Gamma\). The problem has been investigated for the stochastic wave equation by R. C. Dalang and N. E. Frangos [Ann. Probab. 26, No. 1, 187-212 (1998; Zbl 0938.60046)] and also by C. Mueller [ibid. 25, No. 1, 133-151 (1997; Zbl 0884.60054)] when \(d=2\).
In this note the authors treat the general case of arbitrary dimension \(d\) and of arbitrary spatially homogeneous noise for both the stochastic heat and wave equations. Section 2 entitled preliminaries contains heat and wave semigroups, spatially homogeneous Wiener process and questions. Section 3 treates proofs of the results for the case of \(\mathbb{R}^d\). Applications are discussed in Section 4. Extensions to the \(d\)-dimensional torus are contained in Section 5. Section 6 contains two conjectures and some partial answers. For other details see the authors comprehensive references.
For the entire collection see [Zbl 0968.00044].

60H15 Stochastic partial differential equations (aspects of stochastic analysis)