A Solution Theory for a General Class of SPDEs

Abstract

In this article we present a way of treating stochastic partial differential equations with multiplicative noise by rewriting them as stochastically perturbed evolutionary equations in the sense of \cite{picardbook}, where a general solution theory for deterministic evolutionary equations has been developed. This allows us to present a unified solution theory for a general class of SPDEs which we believe has great potential for further generalizations. We will show that many standard stochastic PDEs fit into this class as well as many other SPDEs such as the stochastic Maxwell equation and time-fractional stochastic PDEs with multiplicative noise on sub-domains of $\Rd$. The approach is in spirit similar to the approach in \cite{dapratozabczyk}, but complementing it in the sense that it does not involve semi-group theory and allows for an effective treatment of coupled systems of SPDEs. In particular, the existence of a (regular) fundamental solution or Green's function is not required.