The Obstacle Problem for Quasilinear Stochastic PDEs with Neumann boundary condition

TL;DR

This paper establishes the existence and uniqueness of solutions for the obstacle problem in quasilinear stochastic PDEs with Neumann boundary conditions, using techniques from parabolic potential theory.

Contribution

It introduces a novel approach to solving the obstacle problem for quasilinear SPDEs with Neumann boundary conditions, combining stochastic analysis and potential theory.

Findings

Proves existence and uniqueness of solutions for the obstacle problem.

Characterizes solutions as pairs (u, ν) with specific properties.

Employs parabolic potential theory techniques in stochastic PDE context.

Abstract

We prove the existence and uniqueness of solution of the obstacle problem for quasilinear stochastic partial differential equations (OSPDEs for short) with Neumann boundary condition. Our method is based on the analytical technics coming from 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.