First order equation on random measures as superposition of weak solutions to the McKean-Vlasov equation

The goal of this paper is to define an evolution equation for a curve of random probability measures $(M_t{)}_{t\in [0,T]}\subset \mathcal{P}(\mathcal{P}({\mathbb{R}}^d))$ associated to a non-local drift $b:[0,T]\times {\mathbb{R}}^d\times \mathcal{P}({\mathbb{R}}^d)\to {\mathbb{R}}^d$ and a non-local diffusion term $a:[0,T]\times {\mathbb{R}}^d\times \mathcal{P}({\mathbb{R}}^d)\to {\mathrm{Sym}}_{+}({\mathbb{R}}^{d\times d})$. Then, we show that any solution to that equation can be lifted to a superposition of solutions to a non-linear Kolmogorov-Fokker-Planck equation and also to a superposition of weak solutions to the McKean-Vlasov equations. Finally, we use this superposition result to show how existence and uniqueness can be transferred from the equation on random measures to the associated non-linear Kolmogorov-Fokker-Planck equation and to the McKean-Vlasov equation, assuming uniqueness of the linearized KFP.