A Solution Theory for a General Class of SPDEs

TL;DR

This paper develops a unified solution theory for a broad class of SPDEs with multiplicative noise by reformulating them as perturbed evolutionary equations, avoiding semi-group theory and enabling analysis of complex coupled systems.

Contribution

It introduces a general solution framework for SPDEs that encompasses many standard and complex equations, broadening applicability without relying on Green's functions.

Findings

Unified solution theory for a wide class of SPDEs.

Applicable to stochastic Maxwell and fractional SPDEs.

Does not require fundamental solutions or Green's functions.

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.