Fluctuation for mean field limits of particle systems driven by fBm

Consider a system of $N$ particles, subject to a mean-field type pairwise interaction kernel $K$, each driven by an independent fractional Brownian motion (idiosyncratic noises). Previous works established that, for a large class of non-Lipschitz, possibly singular kernels, the associated McKean-Vlasov equation is well-posed, and the empirical measure converges to its law as $N\to\infty$, with rate of order $N^{-1/2}$ in suitable negative Sobolev norms. In this talk I will present results concerning the Gaussian fluctuations underlying this mean field convergence, validating the optimality of this rate; they are valid for both first order interactions and for kinetic systems. The proofs are based on the use of Girsanov transform and the method of U-statistics first introduced by Sznitman.

Based on ongoing joint work with Avi Mayorcas (Bath) and Johanna Weinberger (Technion).