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 is an element of L 1 boolean AND L infinity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in L<^>1\cap L<^>\infty $$\end{document} . 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.