Absolute continuity of Wasserstein barycenters on manifolds with a lower Ricci curvature bound

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] .