On the Well-posedness of a Class of Non-Autonomous SPDEs: An Operator-Theoretical Perspective

TL;DR

This paper investigates the solvability of non-autonomous stochastic partial differential equations using an operator-theoretical approach, extending previous results to more general, mixed-type equations with stochastic integrals.

Contribution

It introduces a novel operator-theoretical framework for analyzing non-autonomous SPDEs, including equations with mixed types where classical methods are inadequate.

Findings

Extended solvability results for non-autonomous SPDEs.

Applicable to equations with mixed types and stochastic integrals.

Generalizes previous autonomous, linear, white noise cases.

Abstract

We further elaborate on the solvability of stochastic partial differential equations (SPDEs). We shall discuss non-autonomous partial differential equations with an abstract realization of the stochastic integral on the right-hand side. Our approach allows the treatment of equations with mixed type, where classical solution strategies fail to work. The approach extends prior observations in [S\"u\ss, A. \& Waurick, M. A Solution Theory for a General Class of SPDEs. \emph{Stochastics and Partial Differential Equations: Analysis and Computations}, 2017, 5, 278-318], where the respective results were obtained for linear autonomous equations and (multiplicative) white noise.