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 [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.

We examine a 2-dimensional ODE which exhibits explosion in finite time. Considered as an SDE with additive white noise, it is known to be complete -in the sense that for each initial condition there is almost surely no explosion. Furthermore, the associated Markov process even admits an invariant probability measure. On the other hand, as we will show, the corresponding local stochastic flow will almost surely not be strongly complete, i.e. there exist (random) initial conditions for which the solutions explode in finite time.

An interacting particle system with long range interaction is considered. Particles, in addition to the interaction, proliferate with a rate depending on the empirical measure. We prove convergence of the empirical measure to the solution of a parabolic equation with non-local nonlinear transport term and proliferation term of logistic type. T a,N 1 := lim t↑T a,N 1 X a,N t , we impose T a,N 0 = T (a,−),N 1 and X a,N T a,N 0 = X (a,−),N T (a,−),N 1 . Denote by A N the set of all labels a and 2010 Mathematics Subject Classification. Primary: 60K35, 35K57; Secondary: 60F17, 35K58.

We consider a 2-dimensional stochastic differential equation in polar coordinates depending on several parameters. We show that if these parameters belong to a specific regime then the deterministic system explodes in finite time, but the random dynamical system corresponding to the stochastic equation is not only strongly complete but even admits a random attractor.

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