Asymptotic Behavior of a System of Stochastic Particles Subject to Nonlocal Interactions

In this article, we present a rigorous mathematical derivation of a macroscopic model of aggregation, scaling up from a microscopic description of a family of individuals subject to aggregation/repulsion, described by a system of Ito type stochastic differential equations. We analyze the asymptotics of the system for both a large number of particles on a bounded time interval, and its long time behavior, for a fixed number of particles. As far as this second part is concerned, we show that a suitable localizing potential is required, in order that the system may admit a nontrivial invariant distribution.