Long Time $\W_0$-$\widetilde{\W}_1$ type Propagation of Chaos for Mean Field Interacting Particle System

In this paper, a general result on the long time $\W_0$-$\widetilde{\W}_1$ type propagation of chaos, propagation of chaos with regularization effect, for mean field interacting particle system driven by L\'{e}vy noise is derived, where $\W_0$ is one half of the total variation distance while $\widetilde{\W}_1$ 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 $\widetilde{\W}_1$.