On Solutions to Graphon McKean-Vlasov SDEs of Nemytskii-type

Abstract

We study an uncountable system of McKean-Vlasov SDEs with coefficients of Nemytskii-type which are driven by a family of essentially pairwise independent Wiener processes. These SDEs interact through a Graphon kernel by means of their one-dimensional time marginal law densities evaluated in the spatial coordinate. We prove the existence and uniqueness of probabilistically weak solutions to such SDEs under mild conditions on the drift coefficient and the Graphon kernel. Furthermore, we prove that these solutions are, in fact, probabilistically strong by essentially proving a (restricted) Yamada--Watanabe theorem on Fubini extension spaces. For the associated system of nonlinear Fokker-Planck equations, we prove a new uniqueness result where we can allow for density dependent diffusion coefficients of Nemytskii-type.