Séminaire du Pôle Probabilités

In this joint work with Alvin Tse, we prove a generalised version of the central limit theorem for
nonlinear functionals of the empirical measure of i.i.d. random variables, provided
that the functional satisfies some regularity assumptions for the associated linear
functional derivative. We use this result to deal with the contribution of the initialisation
in the convergence of the fluctuations between the empirical measure of interacting diffusions and their
mean-field limiting measure, when the dependence on the measure is nonlinear. A complementary contribution related to the time evolution is treated using the master equation, a parabolic PDE involving L-derivatives with respect to the measure component, which is a stronger notion of derivative that is nonetheless related to the linear functional derivative.