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 explicitly on time and space, for which it is known that there exists a sufficiently Sobolev-regular Schwartz-distributional solution u∈L1∩L∞\\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.