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 $L^1$, and, in the one-dimensional setting, the classical porous medium equation with initial datum in an arbitrary point $x\in\mathbb{R}$. 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 $u$.