Journal of Mathematics of Kyoto University

Space-time regularity for linear stochastic evolution equations driven by spatially homogeneous noise

Abstract

In this paper we study space-time regularity of solutions of the following linear stochastic evolution equation in $\mathcal{S'}(\mathbb{R}^{d})$, the space of tempered distributions on $\mathbb{R}^{d}$: $\begin{array}{cc} (*) & \begin{array}{ll}du(t)=Au(t)dt+dW(t),& t \geqslant 0,\\ u(0)=0.&\end{array} \end{array}$ Here A is a pseudodifferential operator on $\mathcal{S'} (\mathbb{R}^{d})$ whose symbol $q : \mathbb{R}^{d} \to \mathbb{C}$ is symmetric and bounded above, and $\{ W(t)\}_{t\geqslant 0}$ is a spatially homogeneous Wiener process with spectral measure $\mu$. We prove that for any $p \in [1,\infty )$ and any nonnegative weight function $\rho \in L_{\mathrm{loc}}^{1}(\mathbb{R}^{d})$, the following assertions are equivalent:

(1) The problem (*) admits a unique $L^{p}(\rho )$-valued solution;

(2) The weight $\rho$ is integrable and $\int_{\mathbb{R}^{d}}\frac{1}{C-\mathrm{Re}q(\xi )}d\mu (\xi )<\infty$ for sufficiently large $C$.

Under stronger integrability assumptions we prove that the $L^{p}(\rho )$-valued solution has a continuous, resp. Hölder continuous version.

Article information

Source
J. Math. Kyoto Univ., Volume 43, Number 2 (2003), 261-303.

Dates
First available in Project Euclid: 14 August 2009

https://projecteuclid.org/euclid.kjm/1250283728

Digital Object Identifier
doi:10.1215/kjm/1250283728

Mathematical Reviews number (MathSciNet)
MR2051026

Zentralblatt MATH identifier
1056.60057

Citation

Brzeźniak, Zdzisław; van Neerven, Jan. Space-time regularity for linear stochastic evolution equations driven by spatially homogeneous noise. J. Math. Kyoto Univ. 43 (2003), no. 2, 261--303. doi:10.1215/kjm/1250283728. https://projecteuclid.org/euclid.kjm/1250283728