Weak solution for Stochastic Degasperis-Procesi Equation

TL;DR

This paper establishes the existence of weak solutions for the stochastic Degasperis-Procesi equation with multiplicative noise, using kinetic theory and stochastic compactness, and proves pathwise uniqueness leading to strong solutions.

Contribution

It introduces a novel method combining kinetic theory and stochastic compactness to prove existence and uniqueness of solutions for the stochastic Degasperis-Procesi equation.

Findings

Existence of weak kinetic martingale solutions.

Pathwise uniqueness of solutions.

Solutions accommodate physical phenomena like peakons and wave breaking.

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.