Special Session 19: Stochastic Partial Differential Equations

Strong solutions to McKean-Vlasov SDEs with coefficients of Nemytskii-type

Sebastian Grube
Bielefeld University
We study a large class of McKean--Vlasov SDEs with drift and diffusion coefficient depending on the density of the solution`s time marginal laws in a Nemytskii-type of way. A McKean--Vlasov SDE of this kind arises from the study of the associated nonlinear FPKE, for which is known that there exists a bounded Sobolev-regular Schwartz-distributional solution $u$. Via the superposition principle, it is already known that there exists a weak solution to the McKean--Vlasov SDE with time marginal densities $u$. We show that there exists a strong solution the McKean--Vlasov SDE, which is unique among weak solutions with time marginal densities $u$. The main tool is a restricted Yamada--Watanabe theorem for SDEs, which is obtained by an observation in the proof of the classical Yamada--Watanabe theorem.