Hydrodynamic limit and Propagation of Chaos for Brownian Particles reflecting from a Newtonian barrier

TL;DR

This paper extends a stochastic model of particle interaction to multiple particles, establishing a hydrodynamic limit described by a PDE with free boundary conditions and demonstrating existence, uniqueness, and approximation methods.

Contribution

It introduces a multi-particle model reflecting from a Newtonian barrier, proving propagation of chaos and characterizing the hydrodynamic limit with new stochastic and PDE techniques.

Findings

Proved propagation of chaos for the multi-particle system

Characterized the hydrodynamic limit as a PDE with free boundary

Developed an algorithm for approximating the free boundary solution

Abstract

In 2001, Knight constructed a stochastic process modeling the one dimensional interaction of two particles, one being Newtonian in the sense that it obeys Newton's laws of motion, and the other particle being Brownian. We construct a multi-particle analog, using Skorohod map estimates in proving a propagation of chaos and characterizing the hydrodynamic limit as the solution to a PDE with free boundary condition. Stochastic methods are used to show existence and uniqueness for the free boundary problem, and also present an algorithm of approximating the solution.