Uniform-in-time quantitative fluctuations of large scale interacting particle systems

Abstract

We study fluctuations of mean-field interacting particle systems around their McKean--Vlasov limit. Our main result provides a uniform-in-time quantitative central limit theorem for the fluctuation process, with convergence rate of order $N^{-1/2}$ to the corresponding Gaussian limit in the Wasserstein metric. The proof relies on two main ingredients. First, we establish a uniform-in-time weak expansion for specific functionals of the empirical measure around their limiting behavior. This yields, in particular, uniform-in-time control of the convergence of the prelimit variance to its limiting counterpart. We also derive a backward PDE representation of the limiting variance, which is of independent interest. Second, we use Malliavin calculus tools and, in particular, a second-order Poincaré inequality that bounds the Wasserstein distance between the fluctuation process and its Gaussian limit in terms of the first- and second-order Malliavin derivatives of the particle flow. The quantitative convergence rates then follow from a delicate analysis of these derivatives, yielding the sharp estimates required for uniform-in-time control.