Uniform convergence of proliferating particles to the FKPP equation

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 Oelschläger [16], [17]. 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.