On pointwise Malliavin differentiability of solutions to semilinear parabolic SPDEs

TL;DR

This paper establishes pointwise Malliavin differentiability estimates for solutions to a class of semilinear parabolic SPDEs driven by multiplicative noise, advancing the understanding of their regularity properties.

Contribution

It provides new estimates on the first-order Malliavin derivatives of solutions to semilinear parabolic SPDEs with multiplicative noise and polynomial growth nonlinearities.

Findings

Derived pointwise Malliavin derivative estimates for solutions

Extended regularity results to nonlinear SPDEs with polynomial growth

Utilized monotonicity and comparison principles in proofs

Abstract

We obtain estimates on the first-order Malliavin derivative of mild solutions, evaluated at fixed points in time and space, to a class of parabolic dissipative stochastic PDEs on bounded domain of $\mathbb{R}^d$. In particular, such equations are driven by multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a locally Lipschitz continuous function satisfying suitable polynomial growth bounds. The main arguments rely on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces, monotonicity, and a comparison principle.