The main goal of the talk will be to study the geometric structure of the Wasserstein space of random measures $\mathcal{P}_p(\mathcal{P_p(\mathbb{R}^d)})$. In particular, we will characterize absolutely continuous curves of random measures as curves that solves an abstract continuity equation. As a consequence, we recover a Benamou--Brenier-like formula and a nice definition of tangent space, as shown for the classic Wasserstein space in the book by Ambrosio--Gigli--Savaré.

In the first part of the talk, I will recall the main steps of these proofs in the classic case, relying on two superposition principles: the first due to S. Lisini (2006) and the second one to L. Ambrosio (2004). These results will be also fundamental to provide suitable superposition principles for the random measures case, which will lead to the aforementioned results.

The talk is based on a joint work with Giuseppe Savaré.