Strong solutions to McKean-Vlasov SDEs with coefficients of Nemytskii-type: the time-dependent case

TL;DR

This paper establishes the existence and uniqueness of strong solutions for McKean-Vlasov SDEs with Nemytskii-type coefficients, linking solutions to the associated nonlinear Fokker-Planck equations with explicit time-dependent densities.

Contribution

It proves the existence of unique strong solutions for a broad class of McKean-Vlasov SDEs with time-dependent Nemytskii-type coefficients, connecting weak solutions to functional representations.

Findings

Existence and uniqueness of strong solutions for the SDEs.

Weak solutions can be represented as functionals of Brownian motion.

Any Brownian motion plugged into the functional yields a weak solution.

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 distributional solution u in L^1 and L^\infty. 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.