## Beyond propagation of chaos : correlations control in mean-field systems

This talk focuses on N-particle systems with mean-field interactions, following a Brownian or Langevin dynamics. Those corresponds to the underlying system leading to the McKean-Vlasov or Vlasov-Fokker-Planck equations, respectively. In the case of a smooth interaction potential, we derive uniform in time estimates for the many-particle correlation function, with quantified size in N that fits the physical prediction. Our main tools are linear derivatives with respect to the measure, as introduced by Otto and Lions, and in particular Lions expansions in the context of interacting particle systems, as well Glauber calculus, and new ergodic estimates for kinetic equations to treat the Langevin dynamics. Those controls on the correlations are a first step towards the justification of a uniform in time CLT, of a large deviation principle, as well as the derivation of a Bogolyubov correction to the mean-field limit. Joint work with Mitia Duerinckx (ULB).

Area: CS45 - Mckean-Vlasov equations and related PDEs (Elena Issoglio and Stefano Pagliarani)

Keywords: McKean-Vlasov equation, Vlasov-Fokker-Planck equation, correlations control, propagation of chaos, central limit theorem, Lions calculus

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