The Obstacle Problem for Quasilinear Stochastic PDEs with non-homogeneous operator

TL;DR

This paper establishes the existence, uniqueness, and maximum principle for solutions to the obstacle problem in quasilinear stochastic PDEs with non-homogeneous operators, using advanced analytical techniques from potential theory.

Contribution

It introduces a novel approach combining parabolic potential theory and stochastic analysis to solve the obstacle problem for a class of quasilinear SPDEs with non-homogeneous operators.

Findings

Proved existence and uniqueness of solutions.

Established a maximum principle for local solutions.

Expressed solutions as pairs involving a predictable process and a regular measure.

Abstract

We prove the existence and uniqueness of solution of the obstacle problem for quasilinear Stochastic PDEs with non-homogeneous second order operator. Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair $(u,\nu)$ where $u$ is a predictable continuous process which takes values in a proper Sobolev space and $\nu$ is a random regular measure satisfying minimal Skohorod condition. Moreover, we establish a maximum principle for local solutions of such class of stochastic PDEs. The proofs are based on a version of It\^o's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.