Optimal Regularity for Semilinear Stochastic Partial Differential Equations with Multiplicative Noise

TL;DR

This paper establishes the optimal spatial and temporal regularity of solutions to semilinear stochastic PDEs with multiplicative noise, matching the regularity of the stochastic convolution, using elementary methods.

Contribution

It demonstrates that the mild solution shares the same optimal regularity as the stochastic convolution under Lipschitz conditions, with a simple proof approach.

Findings

Mild solutions have optimal regularity properties.

Solutions exhibit Hölder continuity with non-optimal exponents.

Regularity results align with those of stochastic convolutions.

Abstract

This paper deals with the spatial and temporal regularity of the unique Hilbert space valued mild solution to a semilinear stochastic partial differential equation with nonlinear terms that satisfy global Lipschitz conditions. It is shown that the mild solution has the same optimal regularity properties as the stochastic convolution. The proof is elementary and makes use of existing results on the regularity of the solution, in particular, the H\"older continuity with a non-optimal exponent.