Strong solutions to McKean–Vlasov SDEs associated to a class of degenerate Fokker–Planck equations with coefficients of Nemytskii-type

Abstract While the nondegenerate case is well known, there are only few results on the existence of strong solutions to McKean–Vlasov SDEs with coefficients of Nemytskii-type in the degenerate case. We consider a broad class of degenerate nonlinear Fokker–Planck(–Kolmogorov) equations with coefficients of Nemytskii-type. This includes, in particular, the classical porous medium equation perturbed by a first-order term with initial datum in a subset of probability densities, which is dense with respect to the topology inherited from $$L^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> , and, in the one-dimensional setting, the classical porous medium equation with initial datum in an arbitrary point $$x_0\in {\mathbb {R}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> . For these kind of equations the existence of a Schwartz-distributional solution u is well-known. We show that there exists a unique strong solution to the associated degenerate McKean–Vlasov SDE with time marginal law densities u . In particular, every weak solution to 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 yields a weak solution with time marginal law densities u .