The Dirichlet problem for stochastic partial differential equations with nonlocal operators in $C^{1,\sigma}$ open sets

TL;DR

This paper develops a Sobolev regularity theory for stochastic PDEs with nonlocal operators in smooth open sets, establishing existence, uniqueness, and regularity of solutions under broad conditions.

Contribution

It introduces a comprehensive framework for analyzing stochastic PDEs with nonlocal operators in $C^{1,\sigma}$ domains, including existence, uniqueness, and regularity results.

Findings

Existence and uniqueness of strong solutions in weighted Sobolev spaces.

Maximal $L_p$-regularity estimates for solutions.

Applicability to large classes of nonlocal operators and Gaussian noise.

Abstract

This paper provides a comprehensive Sobolev regularity theory for the Dirichlet problem of stochastic partial differential equations in $C^{1,\sigma}$ open sets. We consider substantially large classes of nonlocal operators and generalized Gaussian noise. Our main results include the existence and uniqueness of strong solutions in weighted Sobolev spaces, along with maximal $L_p$-regularity estimates for the solutions.