Particles approximations of Vlasov equations with singular forces : Propagation of chaos

Abstract

We obtain the mean field limit and the propagation of chaos for a system of particles interacting with a singular interaction force of the type $1/|x{|}^{\alpha }$, with $\alpha <1$ in dimension $d\ge 3$. We also provide results for forces with singularity up to $\alpha <d-1$ but with large enough cut-off. This last result thus almost includes the most interesting case of Coulombian or gravitational interaction, but it is also interesting when the strength of the singularity $\alpha$ is larger but close to one, in which case it allows for very small cut-off.

47 pages. This version: improvement of the presentation, in particular of the deterministic results

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