Entropy on the path space and application to singular diffusions and mean-field models

Abstract: In this paper we intend to present a unified treatment of a variety of singular interacting particle systems and their McKean-Vlasov limits. This unified approach is based on the use of the relative entropy on the path space in the spirit of our previous works together with C. L{\'e}onard. We show how it can be used to derive existence and uniqueness for some singular diffusions, in particular linear mean field stochastic particle systems and non linear SDE of McKean-Vlasov type, including $\mathbf L^p-\mathbf L^q$ models, the 2D vortex model associated to the 2D Navier-Stokes equation, sub-Coulombic interactions models or the Patlak-Keller-Segel model. We also show the convergence and propagation of chaos as the number of particles grows to infinity. This is (mainly) obtained at the process level, not only at the Liouville equation (marginals flow) level. The paper thus contains new proofs and extensions of known results, as well as new results.The main results are given at the end of the Introduction.