Convergence rate for Fluctuations of mean field interacting diffusion and application to 2D viscous Vortex model and Coulomb potential

Abstract

For a system of mean field interacting diffusion on ${\mathbb{T}}^d$, the empirical measure ${\mu }^N$ converges to the solution $\mu$ of the Fokker-Planck equation. Refining this mean field limit as a Central Limit Theorem, the fluctuation process ${\rho }_t^N=\sqrt{N}({\mu }_t^N-{\mu }_t)$ convergences to the solution $\rho$ of a linear stochastic PDE on the negative Sobolev space $H^{-\lambda -2}({\mathbb{T}}^d)$. The main result of the paper is to establish a rate for such convergence: we show that $|\mathbb{E}[\Phi ({\rho }_t^N)]-\mathbb{E}[\Phi ({\rho }_t)]|=\mathcal{O}(\frac{1}{\sqrt{N}})$, for smooth functions on $H^{-\lambda -2}({\mathbb{T}}^d)$. The strategy relies on studying the generators of the processes ${\rho }^N$ and $\rho$ on $H^{-\lambda -2}({\mathbb{T}}^d)$, and thus estimating their difference. Among others, this requires to approximate in probability $\rho$ with solutions to stochastic diffential equations on the Hilbert space $H^{-\lambda -2}({\mathbb{T}}^d)$. The flexibility of the approach permits to establish a rate for the fluctuations, not only in case of a regular drift, but also for the the 2D viscous Vortex model, governed by the Biot-Savart kernel, and for the repulsive Coulomb potential.