The KPP equation as a scaling limit of locally interacting Brownian particles

TL;DR

This paper demonstrates that the Fisher-KPP equation can be derived as a scaling limit of a system of Brownian particles with local interactions, addressing challenges in controlling local concentrations unlike in mean field models.

Contribution

It introduces a novel approach to prove the scaling limit of locally interacting Brownian particles converging to the Fisher-KPP equation, extending previous methods to local interaction regimes.

Findings

Fisher-KPP equation is the scaling limit of locally interacting Brownian particles.

Overcomes difficulties in controlling local particle concentrations.

Adapts methods from mean free path models to local interaction settings.

Abstract

Fisher-KPP equation is proved to be the scaling limit of a system of Brownian particles with local interaction. Particles proliferate and die depending on the local concentration of other particles. Opposite to discrete models, controlling concentration of particles is a major difficulty in Brownian particle interaction; local interactions instead of mean field or moderate ones makes it more difficult to implement the law of large numbers properties. The approach taken here to overcome these difficulties is largely inspired by A. Hammond and F. Rezakhanlou [10] implemented there in the mean free path case instead of the local interaction regime.