A Hilbertian approach for fluctuations on the McKean-Vlasov model

Abstract 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 ), 2 D 0 and converge in C ([0, T ], W −(2+2 D ), D 0 ) to a Ornstein-Uhlenbeck process obtained as the solution of a Langevin equation in W −(4+2 D ), D 0 , where D is equal to 1 + [ d 2 ] . It appears in the proofs that the spaces W −(1 → D ), 2 D 0 and W −(2−2 D ), D 0 are minimal Sobolev spaces in which to immerse the fluctuations, which was our aim following a physical point of view.