A study of the metric measure space of probability measures via a purely atomic superposition principle

TL;DR

This paper investigates the structure of probability measures on metric spaces using a superposition principle, revealing atomic properties and geometric inequalities in Wasserstein spaces over Euclidean and Riemannian manifolds.

Contribution

It introduces a purely atomic superposition principle for the continuity equation on metric measure spaces of probability measures, extending results from Euclidean spaces to Riemannian manifolds.

Findings

Atomic property is inherited by lifted curves under certain conditions.

The analysis extends to Riemannian manifolds via Nash embedding theorem.

The Wasserstein space with a suitable measure satisfies a Bakry-Émery curvature condition.

Abstract

We study the continuity equation on the metric measure space $(\mathcal{P}_p(X),W_p,Q)$, when $X$ is either the Euclidean space or a compact, oriented, and boundaryless Riemannian manifold, for some suitable reference measure $Q \in \mathcal{P}_p(\mathcal{P}_p(X))$, which by construction are concentrated over purely atomic measures. In fact, we consider the equation $\partial_t M_t +\operatorname{div}_{\mathcal{P}}(b_t M_t) = 0$, where $(M_t)_{t\in[0,T]} \subset\mathcal{P}(\mathcal{P}(X))$ and $b:[0,T]\times X \times \mathcal{P}(X) \to TX$, assuming that $M_t\ll Q$ for all $t\in[0,T]$, to then show when the purely atomic property is inherited by the liftings of the curve $M_t$ given by the nested superposition principle. On the Euclidean space, the main assumption is that the $r$-capacity of the diagonal $\Delta \subset \mathbb{R}^d \times \mathbb{R}^d$ is zero with respect to $\nu \otimes \nu$, where $\nu$ is the barycenter of the reference measure $Q$. We will give sufficient conditions to ensure it, and in particular, thanks to the Nash embedding theorem, this analysis will allow us to extent the main results from the Euclidean space to Riemannian manifolds. Finally, we exploit this atomic superposition principle to show the lack of the Sobolev-to-Lipschitz property and the Poincar\'e inequality for functions in $W^{1,p}(\mathcal{P}_p(X),W_p,Q)$. Then, we complete the analysis showing that, however, the $L^2$-Wasserstein space endowed with suitable reference measure $Q$, satisfies a Bakry--\'Emery curvature condition.