Long Time Entropy-Cost type Propagation of Chaos

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. The result shows that the Kac's chaotic property measured in relative entropy at any positive time can only depend on the weaker initial one measured in $L^2$-Wasserstein distance. Moreover, under dissipative assumption, the long time entropy-cost type propagation of chaos can also be captured. The results are also available in path dependent case, where the log-Sobolev inequality for McKean-Vlasov SDEs does not hold.