  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.
