Alessandro Pinzi est doctorant l'Université Luigi Bocconi.
Titre : The Wasserstein geometry of random measures through nested superposition principle
Résumé : In this talk, we will mainly work with the space of (laws of) random measures $\mathcal{P}(\mathcal{P}(\mathbb{R}^d))$. We will introduce suitable first order evolution equations for random measures, for which we will prove a nested superposition principle, in the spirit of the results by Ambrosio-Figalli-Trevisan.
As an application, we study the geometric structure of the space of random measures $\mathcal{P}_p (\mathcal{P}_p(\mathbb{R}^d))$, endowed with the Wasserstein-on-Wasserstein metric, in analogy with the classic Wasserstein space $\mathcal{P}_p(\mathbb{R}^d)$ shown in the book by Ambrosio-Gigli-Savaré. Indeed, a specific form for the previously introduced equations is exactly the natural continuity equation for laws of random measures. Then, the nested superposition principle allows us to: characterize the absolutely continuous curves on the Wasserstein-on-Wasserstein space as solutions of some continuity equation; define and characterize its tangent bundle; and prove a Benamou-Brenier-like formula.
The talk is partially based on a joint work with Giuseppe Savaré.