On the differentiability of solutions to singularly perturbed SPDEs

TL;DR

This paper investigates how solutions to certain singularly perturbed stochastic PDEs depend smoothly on a small parameter, even when the perturbation operator is singular, advancing understanding of solution regularity in complex stochastic systems.

Contribution

It establishes the differentiability of mild solutions with respect to the perturbation parameter in semilinear stochastic evolution equations with singular operators.

Findings

Proves differentiability of solutions in the presence of singular perturbations.

Extends existing theory to operators with non-overlapping domains.

Provides a framework for analyzing sensitivity in stochastic PDEs with singular perturbations.

Abstract

We consider semilinear stochastic evolution equations on Hilbert spaces with multiplicative Wiener noise and linear drift term of the type $A + \varepsilon G$, with $A$ and $G$ maximal monotone operators and $\varepsilon$ a "small" parameter, and study the differentiability of mild solutions with respect to $\varepsilon$. The operator $G$ can be a singular perturbation of $A$, in the sense that its domain can be strictly contained in the domain of $A$.