  In this paper, the quantitative propagation of chaos for mean field
interacting particle system with multiplicative noise in $L^2$-Wasserstein
distance is derived. When the diffusion is distribution free, the quantitative
propagation of chaos in relative entropy as well as total variation distance is
also obtained, where the initial distribution of mean field interacting
particles is allowed to be singular with that of the limit equation.
Furthermore, a general result on the long time quantitative propagation of
chaos in relative entropy as well as in $L^2$-Wasserstein distance is provided
and is applied in (path dependent) mean field interacting particle system as
well as stochastic Hamiltonian system under dissipative condition.
