Differential Inclusions in Wasserstein Spaces: The Cauchy-Lipschitz Framework

In this article, we propose a general framework for the study of differential in the Wasserstein space of probability measures. Based on earlier insights on the structure of continuity equations, we define of differential inclusions as absolutely continuous curves whose velocity fields are measurable selections of multifunction taking their in the space of vector fields. In this general setting, we prove three the founding results of the theory of differential inclusions: Filippov's, the Relaxation theorem, and the compactness of the solution sets. contributions -- which are based on novel estimates on solutions of equations -- are then applied to derive a new existence result for non-linear mean-field optimal control problems with closed-loop controls.