Fokker-Planck-Kolmogorov inclusions of the mean field type

The paper studies a dynamical system in the Wasserstein space, where the evolution is governed by a Fokker-Planck-Kolmogorov equation with coefficients chosen from a prescribed set that depends at each point on that point and on the current measure.

Under the assumption that the multivalued mapping determining the constraints is convex-valued, upper semicontinuous and satisfies certain growth conditions, we prove an existence theorem and establish the compactness of the solution set.

Moreover, we study an optimal control problem for such dynamical systems.