A Hilbertian approach for fluctuations on the McKean-Vlasov model

We consider the sequence of fluctuation processes associated with the empirical measures of the interacting particle system approximating the d-dimensional McKean-Vlasov equation and prove that they are tight as continuous processes with values in a precise weighted Sobolev space. More precisely, we prove that these fluctuations belong uniformly (with respect to the size of the system and to time) to W−(1+D), 2D0 and converge in C([0, T], W−(2+2D), D0) to a Ornstein-Uhlenbeck process obtained as the solution of a Langevin equation in W−(4+2D), D0, where D is equal to 1 + [d2]. It appears in the proofs that the spaces W−(1 → D), 2D0 and W−(2−2D), D0 are minimal Sobolev spaces in which to immerse the fluctuations, which was our aim following a physical point of view.