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We consider the fully-coupled McKean-Vlasov equation with multi-time-scalepotentials, and all the coefficients depend on the distributions of both theslow component and the fast motion. By studying the smoothness of the solutionof the non-linear Poisson equation on Wasserstein space, we derive theasymptotic limit as well as the quantitative error estimate of the convergencefor the slow process. Extra homogenized drift term containing derivative in themeasure argument of the solution of the Poisson equation appears in the limit,which seems to be new and is unique for systems involving the fastdistribution.

Poisson equation on Wasserstein space and diffusion approximations for McKean-Vlasov equation