Probabilistic representation of a class of non conservative nonlinear Partial Differential Equations

TL;DR

This paper introduces a new class of nonlinear stochastic differential equations linked to non-conservative nonlinear PDEs, establishing existence, uniqueness, and a particle system approximation with convergence to PDE solutions.

Contribution

It presents a novel stochastic framework for non-conservative nonlinear PDEs, including an interacting particle system and proof of propagation of chaos.

Findings

Existence and uniqueness of solutions under various assumptions

Development of an interacting particle system for approximation

Convergence of the particle system to PDE solutions

Abstract

We introduce a new class of nonlinear Stochastic Differential Equations in the sense of McKean, related to non conservative nonlinear Partial Differential equations (PDEs). We discuss existence and uniqueness pathwise and in law under various assumptions. We propose an original interacting particle system for which we discuss the propagation of chaos. To this system, we associate a random function which is proved to converge to a solution of a regularized version of PDE.