Entropic-Wasserstein barycenters: PDE characterization, regularity and CLT

In this paper, we investigate properties of entropy-penalized Wasserstein barycenters introduced by Bigot, Cazelles and Papadakis (2019) as a regularization of Wasserstein barycenters first presented by Agueh and Carlier (2011). After characterizing these barycenters in terms of a system of Monge-Amp`ere equations, we prove some global moment and Sobolev bounds as well as higher regularity properties. We finally establish a central limit theorem for entropic-Wasserstein barycenters.