Differential inclusions in Wasserstein spaces: The Cauchy-Lipschitz framework

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