Particle approximation of Vlasov equations with singular forces: Propagation of Chaos

We justify the mean field approximation and prove the propagation of chaos for a system of particles interacting with a singular interaction force of the type 1/|x|^α, with α < 1 in dimension d ≥ 3. We also provide results for forces with singularity up to α < d - 1 but with a large enough cut-off. This last result thus almost includes the case of Coulombian or gravitational interactions, but it also allows for a very small cut-off when the strength of the singularity a is larger but close to one.