Correlated Brownian Motions as an Approximation to Deterministic Mean-Field Dynamics

Abstract

We analyze the transition from deterministic mean-field dynamics of several large particles and infinitely many small particles to a stochastic motion of the large particles. In this transition the small particles become the random medium for the large particles, and the motion of the large particles becomes stochastic. Under the assumption that the empirical velocity distribution of the small particles is governed by a probability density ?, the mean-field force can be represented as the negative gradient of a scaled version of ?. The stochastic motion is described by a system of stochastic ordinary differential equations driven by Gaussian space-time white noise and the mean-field force as a shift-invariant integral kernel. The scaling preserves a small parameter in the transition, the so-called correlation length. In this set-up, the separate motion of each particle is a classical Brownian motion (Wiener process), but the joint motion is correlated through the mean-field force and the noise. Therefore, it is not Gaussian. The motion of two particles is analyzed in detail and a diffusion equation is deduced for the difference in the positions of the two particles. The diffusion coefficient in the latter equation is spatially dependent, which allows us to determine regions of attraction and repulsion of the two particles by computing the probability fluxes. The result is consistent with observations in the applied sciences, namely that Brownian particles get attracted to one another if the distance between them is smaller than a critical small parameter. In our case, this parameter is shown to be proportional to the aforementioned correlation length.