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

We consider a large class of nonlinear FPKEs with coefficients of Nemytskii type depending explicitly on time and space, for which it is known that there exists a sufficiently Sobolev-regular Schwartz-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.