  We consider a system of classical Brownian particles interacting via a smooth
long-range potential in the mean-field regime, and we analyze the propagation
of chaos in form of sharp, uniform-in-time estimates on many-particle
correlation functions. Our results cover both the kinetic Langevin setting and
the corresponding overdamped Brownian dynamics. The approach is mainly based on
so-called Lions expansions, which we combine with new diagrammatic tools to
capture many-particle cancellations, as well as with fine ergodic estimates on
the linearized mean-field equation, and with discrete stochastic calculus with
respect to initial data. In the process, we derive some new ergodic estimates
for the linearized Vlasov-Fokker-Planck kinetic equation that are of
independent interest. Our analysis also leads to uniform-in-time concentration
estimates and to a uniform-in-time quantitative central limit theorem for the
empirical measure associated with the particle dynamics.
