Nonlinear parabolic SPDEs involving Dirichlet operators

TL;DR

This paper investigates the existence, uniqueness, and regularity of solutions to nonlinear stochastic PDEs involving Dirichlet operators, using advanced probabilistic and potential theory methods.

Contribution

It introduces a novel approach combining backward doubly stochastic differential equations with Dirichlet form theory for nonlinear SPDEs.

Findings

Established conditions for existence and uniqueness of solutions.

Proved regularity properties of solutions.

Developed a framework integrating stochastic calculus and potential theory.

Abstract

We study the problem of existence, uniqueness and regularity of probabilistic solutions of the Cauchy problem for nonlinear stochastic partial differential equations involving operators corresponding to regular (nonsymmetric) Dirichlet forms. In proofs we combine the methods of backward doubly stochastic differential equations with those of probabilistic potential theory and Dirichlet forms.