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

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 $\rho^N_t= \sqrt{N}( \mu^N_t -\mu_t)$ 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 $|\mathbb{E}[\Phi(\rho_t^N)] - \mathbb{E}[\Phi(\rho_t)]| = \mathcal{O}(\tfrac{1}{\sqrt{N}})$, 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.