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We consider a large class of nonlinear FPKEs with coefficients ofNemytskii-type depending explicitly on time and space, for which it is knownthat there exists a sufficiently Sobolev-regular distributional solution u inL^1 and L^\infty. We show that there exists a unique strong solution to theassociated McKean-Vlasov SDE with time marginal law densities u. In particular,every weak solution of this equation with time marginal law densities u can bewritten as a functional of the driving Brownian motion. Moreover, plugging anyBrownian motion into this very functional produces a weak solution with timemarginal law densities u.

Strong solutions to McKean-Vlasov SDEs with coefficients of Nemytskii-type: the time-dependent case