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.We consider the fully-coupled McKean–Vlasov equation with multi-time-scale potentials, and all the coefficients depend on the distributions of both the slow component and the fast motion. By studying the smoothness of the solution of the Poisson equation on Wasserstein space, we derive the asymptotic limit as well as the quantitative error estimate of the convergence for the slow process. An extra homogenized drift term containing derivative in the measure argument of the solution of the Poisson equation appears in the limit, which seems to be new and is unique for systems involving the fast distribution.KeywordsPoisson equation on Wasserstein spacediffusion approximationMcKean–Vlasov equationmultiscale processesMSC codes60J6060F0535J6070K70
Li et al. (Fri,) studied this question.