Absolute continuity of Wasserstein barycenters on manifolds with a lower Ricci curvature bound
Jianyu Ma
TL;DR
This work proves that Wasserstein barycenters of probability measures on a complete Riemannian manifold with a lower Ricci curvature bound are absolutely continuous when the driving measure assigns mass to AC measures. The authors introduce displacement functionals built from a Hessian equality for barycenters and employ Souslin-space techniques to manage measurability and compactness; these tools enable a noncompact generalization of prior compact-manifold results. A Hessian equality for finite barycenters underpins a second-order calculus that yields entropy-type control along displacement interpolations. The combination of approximation via finite marginals, Hörmander-like Hessian analysis, and Souslin-space compactness culminates in a robust proof that the barycenter inherits absolute continuity, with potential implications for barycentric Jensen-type inequalities and curvature-dimension theory.
Abstract
Given a complete Riemannian manifold $M$ with a lower Ricci curvature bound, we consider barycenters in the Wasserstein space $\mathcal{W}_2(M)$ of probability measures on $M$. We refer to them as Wasserstein barycenters, which by definition are probability measures on $M$. The goal of this article is to present a novel approach to proving their absolute continuity. We introduce a new class of displacement functionals exploiting the Hessian equality for Wasserstein barycenters. To provide suitable instances of such functionals, we revisit Souslin space theory, Dunford-Pettis theorem and the de la Vallée Poussin criterion for uniform integrability. Our method shows that if a probability measure $\mathbb{P}$ on $\mathcal{W}_2(M)$ gives mass to absolutely continuous measures on $M$, then its unique barycenter is also absolutely continuous. This generalizes the previous results on compact manifolds by Kim and Pass arXiv:1412.7726 [math.AP] .