We consider a system of classical particles, interacting via a smooth, long-range potential, in the mean-field regime, and we analyze the propagation of chaos in form of sharp estimates on many-particle correlations. While approaches based on BBGKY hierarchy are doomed by uncontrolled losses of derivatives, we propose a novel non-hierarchical approach relying on discrete stochastic calculus with respect to initial data. This result allows to rigorously truncate the BBGKY hierarchy to an arbitrary precision on the mean-field timescale, thus justifying Bogolyubov corrections to the mean-field Vlasov description. As a by-product, we discuss the justification of the Lenard-Balescu relaxation and of the Landau approximation.

This is partly based on joint works with L. Saint-Raymond, R. Winter, and P.-E. Jabin.