Degenerate parabolic SPDEs

TL;DR

This paper investigates the well-posedness of scalar semilinear degenerate parabolic SPDEs with stochastic forcing across all dimensions, employing kinetic solutions and vanishing viscosity methods.

Contribution

It introduces a framework for analyzing degenerate parabolic SPDEs using kinetic solutions and proves existence via vanishing viscosity, extending understanding in stochastic PDE theory.

Findings

Established well-posedness in any space dimension.

Developed a kinetic solution framework for degenerate SPDEs.

Proved existence of solutions through vanishing viscosity approximation.

Abstract

We study the Cauchy problem for a scalar semilinear degenerate parabolic partial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic solution which is well suited for degenerate parabolic problems and supplies a good technical framework to prove the comparison principle. The proof of existence is based on the vanishing viscosity method: the solution is obtained by a compactness argument as the limit of solutions of nondegenerate approximations.