PUB - Publikationen an der Universität Bielefeld

The mean-field limit of interacting diffusions without exchangeability, caused byweighted interactions and non-i.i.d. initial values, are investigated. The weights could be signedand unbounded. The result applies to a large class of singular kernels including the Biot--Savartlaw. We demonstrate a flexible type of mean-field convergence, in contrast to the typical conver-gence of 1N Sigma(N)(i=1)delta(Xi). More specifically, the sequence of signed empirical measure processes witharbitrary uniform l (R)-weights, r > 1, weakly converges to a coupled PDE's, such as the dynamicsdescribing the passive scalar advected by the 2D Navier--Stokes equation. Our method is based ona tightness/compactness argument and makes use of the systems' uniform Fisher information. Themain difficulty is to determine how to propagate the regularity properties of the limits of empiricalmeasures in the absence of the DeFinetti--Hewitt--Savage theorem for the nonexchangeable case. Tothis end, a sequence of random measures, which merges weakly with a sequence of weighted empiricalmeasures and has uniform Sobolev regularity, is constructed through the disintegration of the jointlaws of particles.