Propagation of Chaos in Relative Entropy with Weak Initial Conditions

Abstract: Due to the regularization effect of the stochastic noise, the quantitative entropy-cost type propagation of chaos for mean field interacting particle system is proposed. Different from the existing results on the propagation of chaos in relative entropy, we replace the finite relative entropy between the initial distribution of interacting particles and that of the limit McKean-Vlasov SDEs by the finite $L^2$-Wasserstein distance, which weakens the initial conditions in some sense. Furthermore, a general result on the long time entropy-cost type propagation of chaos is provided and is applied in several degenerate models, including path dependent as well as kinetic mean field interacting particle system with dissipative coeffcients, where the log-Sobolev inequality for the the distribution of the solution to the limit McKean-Vlasov SDEs does not hold.