Sobolev regularity theory for stochastic reaction-diffusion-advection equations with spatially homogeneous colored noises and infinitesimal generators of subordinate Brownian motions

TL;DR

This paper develops a Sobolev regularity framework for complex stochastic reaction-diffusion-advection equations driven by spatially homogeneous colored noises, introducing a new noise condition and proving solution regularity in mixed norm spaces.

Contribution

It introduces the strongly reinforced Dalang's condition for colored noise and establishes regularity results for solutions in mixed norm spaces, advancing stochastic PDE theory.

Findings

Existence and uniqueness of solutions under new noise condition

Proven space-time Hölder regularity of solutions

Enhanced understanding of nonlinearities and stochastic forces relationship

Abstract

This article investigates the existence, uniqueness, and regularity of solutions to nonlinear stochastic reaction-diffusion-advection equations (SRDAEs) with spatially homogeneous colored noises and infinitesimal generators of subordinate Brownian motions in mixed norm $L_q(L_p)$-spaces. We introduce a new condition (strongly reinforced Dalang's condition) on colored noise, which facilitates a deeper understanding of the complicated relation between nonlinearities and stochastic forces. Additionally, we establish the space-time H\"older type regularity of solutions.