Well-posedness of the Stochastic Degasperis-Procesi Equation

TL;DR

This paper proves the existence of global solutions for the stochastic Degasperis-Procesi equation with additive noise, improving deterministic results by using kinetic theory and initial data in specific function spaces.

Contribution

It establishes the well-posedness of the stochastic Degasperis-Procesi equation with additive noise, extending deterministic solvability results to the stochastic setting.

Findings

Existence of global pathwise solutions for SDP with additive noise.

Improved deterministic solvability results in the zero-noise case.

Application of kinetic theory to stochastic PDEs.

Abstract

This article studies the Stochastic Degasperis-Procesi (SDP) equation on $\mathbb{R}$ with an additive noise. Applying the kinetic theory, and considering the initial conditions in $L^2(\mathbb{R})\cap L^{2+\delta}(\mathbb{R})$, for arbitrary small $\delta>0$, we establish the existence of a global pathwise solution. Restricting to the particular case of zero noise, our result improves the deterministic solvability results that exist in the literature.