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

Abstract: We consider a large class of nonlinear FPKEs with coefficients of Nemytskii-type depending \textit{explicitly} on time and space, for which it is known that there exists a sufficiently Sobolev-regular distributional solution $u\in L^1\cap L^\infty$. We show that there exists a unique strong solution to the associated McKean-Vlasov SDE with time marginal law densities $u$. In particular, every weak solution of this equation with time marginal law densities $u$ can be written as a functional of the driving Brownian motion. Moreover, plugging any Brownian motion into this very functional produces a weak solution with time marginal law densities $u$.