Propagation of Chaos for a Class of First Order Models with Singular Mean Field Interactions

Dynamical systems of N particles in RD interacting by a singular pair potential of mean field type are considered. The systems are assumed to be of gradient type and the existence of a macroscopic limit in the many particle limit is established for a large class of singular interaction potentials in stochastic as well as deterministic settings. The main assumption on the potentials is an appropriate notion of quasi-convexity. When $D=1$ the convergence result is sharp when applied to strongly singular repulsive interactions and for a general dimension D the result applies to attractive interactions with Lipschitz singular interaction potentials, leading to stochastic particle solutions to the corresponding macroscopic aggregation equations. The proof uses the theory of gradient flows in Wasserstein spaces of Ambrosio, Gigli, and Savaré.