Nested superposition principle for random measures and the geometry of the Wasserstein on Wasserstein space

We study the geometric structure of the space of random measures ${\mathcal{P}}_p({\mathcal{P}}_p(X))$, endowed with the Wasserstein on Wasserstein metric, where $(X,d)$ is a complete separable metric space. In this setting, we prove a metric superposition principle, in the spirit of the result by S. Lisini, that will allow us to recover important geometric features of the space. When $X$ is ${\mathbb{R}}^d$, we study the differential structure of ${\mathcal{P}}_p({\mathcal{P}}_p({\mathbb{R}}^d))$ in analogy with the simpler Wasserstein space ${\mathcal{P}}_p({\mathbb{R}}^d)$. We show that continuity equations for random measures involving the abstract concept of derivation acting on cylinder functions can be more conveniently described by suitable non-local vector fields $b:[0,T]\times {\mathbb{R}}^d\times \mathcal{P}({\mathbb{R}}^d)\to {\mathbb{R}}^d$. In this way, we can: 1) characterize the absolutely continuous curves on the Wasserstein on Wasserstein space; 2) define and characterize its tangent bundle; 3) prove a superposition principle for the solutions to the standard non-local continuity equation in terms of solutions of interacting particle systems.