The obstacle problem for quasilinear stochastic PDEs: Analytical approach

TL;DR

This paper establishes existence and uniqueness for quasilinear stochastic PDEs with obstacles using analytical techniques from parabolic potential theory, providing a rigorous framework for such complex equations.

Contribution

It introduces a novel analytical approach to solve quasilinear stochastic PDEs with obstacles, characterizing solutions as pairs involving a predictable process and a regular measure.

Findings

Proves existence and uniqueness of solutions

Characterizes solutions as pairs (u, ν)

Uses parabolic potential theory techniques

Abstract

We prove an existence and uniqueness result for quasilinear Stochastic PDEs with obstacle (OSPDE in short). 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 the minimal Skohorod condition.