Evolution of interacting particles in a Brownian medium

Abstract

Evolution of interacting particles in a random medium is studied by treating their spatial distribution as a measure-valued process. Each individual particle is assumed to move according to a stochastic differential equation, the interaction being manifest both through the correlation of the driving Wiener processes and through the explicit dependence of the "drift" and "diffusion" coefficients on the overall distribution of the particles. The evolution equation for the above-mentioned measure-valued process is derived and the uniqueness of its solutions established using the techniques developed by Bismut and Kunita.