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.