Weak solution for Stochastic Degasperis-Procesi Equation

Abstract

This article is concerned with the existence of solution to the stochastic Degasperis-Procesi equation on $\mathbb{R}$ with an infinite dimensional multiplicative noise and integrable initial data. Writing the equation as a system composed of a stochastic nonlinear conservation law and an elliptic equation, we are able to develop a method based on the conjugation of kinetic theory with stochastic compactness arguments. More precisely, we first apply the stochastic Jakubowski-Skorokhod representation theorem to show the existence of a weak kinetic martingale solution. Next, we prove the pathwise uniqueness and invoke the Yamada-Watanabe-Engelbert theorem to conclude that the solution is strong in the probabilistic sense. In this framework, the solution is a stochastic process with sample paths in Lebesgue spaces, which are compatible with peakons and wave breaking physical phenomenon.

References in corpus (6)

• Multi-peakon solutions of the Degasperis-Procesi equation
• Scalar conservation laws with stochastic forcing
• Degasperis-Procesi peakons and the discrete cubic string
• A New Integrable Equation with Peakon Solutions
• Stochastic Reaction-diffusion Equations Driven by Jump Processes
• Well-posedness of the Stochastic Degasperis-Procesi Equation