Long Time ${\text{W}}_0$-${\tilde{\text{W}}}_1$ type Propagation of Chaos for Mean Field Interacting Particle System

In this paper, a general result on the long time ${\text{W}}_0$-${\tilde{\text{W}}}_1$ type propagation of chaos, propagation of chaos with regularization effect, for mean field interacting particle system driven by Lévy noise is derived, where ${\text{W}}_0$ is one half of the total variation distance while ${\tilde{\text{W}}}_1$ is the $L^1$-Wasserstein distance. By using the method of coupling, the general result is applied to mean field interacting particle system driven by multiplicative Brownian motion and additive $\alpha (\alpha >1)$-stable noise respectively, where the non-interacting drift is assumed to be dissipative in long distance and the initial distribution of interacting particle system converges to that of the limit equation in ${\tilde{\text{W}}}_1$.