The University of Arizona: The Wasserstein geometry of random measures through superposition principles

When $X$ is $\mathbb{R}^d$, we will se also the differential structure of \(\mathcal{P}_p(\mathcal{P}_p(\mathbb{R}^d))\) in analogy with the classic Wasserstein space $\mathcal{P}_p(\mathbb{R}^d)$. We show that continuity equations for laws of random measures involving the abstract concept of derivation acting on cylinder functions can be more conveniently described by suitable non-local vector fields $b:[0,T]\times \mathbb{R}^d \times \mathcal{P}_p(\mathbb{R}^d) \to \mathbb{R}^d$. In this way, we can: characterize the absolutely continuous curves on the Wasserstein-on-Wasserstein space; define and characterize its tangent bundle; prove a Benamou-Brenier-like formula; prove a superposition principle for the solutions to the standard non-local continuity equation in terms of solutions of interacting particle systems. The talk is based on a joint work with Giuseppe Savar\’e.

Presented by Alessandro Pinzi, Bocconi University

A Modeling & Computation Seminar