## Journal of Mathematics of Kyoto University

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

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

Zdzisław Brzeźniak and Jan van Neerven

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/kjm/1250283728

**Mathematical Reviews number (MathSciNet)**

MR2051026

**Zentralblatt MATH identifier**

1056.60057

**Subjects**

Primary: 60H15: Stochastic partial differential equations [See also 35R60]

Secondary: 35B65: Smoothness and regularity of solutions 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 60G15: Gaussian processes 60H05: Stochastic integrals

#### 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