On the Cauchy problem of a degenerate parabolic-hyperbolic PDE with L\'{e}vy noise

TL;DR

This paper studies the impact of Lévy noise on degenerate parabolic PDEs, establishing well-posedness through a weak entropy solution framework and extending classical techniques to stochastic settings.

Contribution

It introduces a novel weak entropy solution framework for stochastic degenerate PDEs with Lévy noise and proves existence and uniqueness using vanishing viscosity and doubling techniques.

Findings

Established well-posedness of stochastic degenerate PDEs with Lévy noise

Developed a weak entropy solution framework for such equations

Extended classical PDE techniques to stochastic noise context

Abstract

In this article we deal with stochastic perturbation of degenerate parabolic partial differential equations (PDEs). The particular emphasise is on analysing the effect of multiplicative L\'{e}vy noise to such problems and establishing wellposedness by developing a suitable weak entropy solution framework. The proof of existence is based on the vanishing viscosity technique. The uniqueness is settled by interpreting Kruzkov's doubling technique in the presence noise.