Uniform convergence of proliferating particles to the FKPP equation

TL;DR

This paper proves that a system of proliferating Brownian particles with local interaction converges uniformly to the FKPP equation, using a novel semigroup approach inspired by SPDE literature.

Contribution

It introduces a new semigroup method to establish uniform convergence of particle systems with local proliferation to the FKPP equation.

Findings

Empirical process converges uniformly in space to FKPP solution.

The approach is inspired by stochastic PDE techniques.

Provides a rigorous link between particle systems and PDEs.

Abstract

In this paper we consider a system of Brownian particles with proliferation whose rate depends on the empirical measure. The dependence is more local than a mean field one and has been called moderate interaction by Oelschlager [17], [18]. We prove that the empirical process converges, uniformly in the space variable, to the solution of the Fisher-Kolmogorov-Petrowskii-Piskunov equation. We use a semigroup approach which is new in the framework of these systems and is inspired by some literature on stochastic partial differential equations.