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

par Jianyu Ma (IMT)

lundi 26 juin 2023, 14:00 → 15:00 Europe/Paris

Salle Cavaillès (1R2-132)

Barycenter is the notion of mean for probability measures on metric spaces. In the Wasserstein space of probability measures over a Riemannian manifold, (Wasserstein) barycenters are themselves probability measures on the manifolds. Assuming that the manifold has lower Ricci curvature bound, we prove that if a probability measure on the Wasserstein space gives mass to absolutely continuous measures, then its barycenter is also absolutely continuous. The case of compact manifolds for the above proposition (with an extra assumption included) was proven by Kim and Pass in 2017. In this talk, the author explains a novel approach to the absolute continuity of Wasserstein barycenters. Especially, we define a class of generalized displacement functionals characterising the absolute continuity. We apply Souslin space theory to find proper instances of such functionals for the given measure on the Wasserstein space. If we still have time, we will present a simpler proof for the case of the real line using calculation.