Space-time regularity for linear stochastic evolution equations driven by spatially homogeneous noise (Q1890279)

Consider the following linear stochastic evolution equation on the space of tempered distributions \({\mathcal S}'\): \[ d u(t) = A u(t) dt + d W(t) \text{ for } t \geq 0, \quad \text{and}\quad u(0)=0, \] where \(A\) is a pseudodifferential operator on \({\mathcal S}'\) with symmetric and bounded above symbol \(q\), and \(\{ W(t)\}_{t\geq 0}\) is a spatially homogeneous Wiener process with spectral measure \(\mu\). The authors prove that, for any \(p\geq 1\) and nonnegative weight function \(\rho \in L^1_{\text{loc}}\), the following assertions are equivalent: (1) The above problem admits a unique \(L^p (\rho)\)-valued solution; (2) the weight \(\rho\) is integrable and, for sufficiently large \(C\), one has: \(\int {{1\over{ C- \Re q (t)}}} d \mu (t) <\infty\). Moreover, under stronger integrability assumptions, the solution has a continuous, even Hölder continuous version.