  This paper is concerned by the study of barycenters for random probability
measures in the Wasserstein space. Using a duality argument, we give a precise
characterization of the population barycenter for various parametric classes of
random probability measures with compact support. In particular, we make a
connection between averaging in the Wasserstein space as introduced in Agueh
and Carlier (2011), and taking the expectation of optimal transport maps with
respect to a fixed reference measure. We also discuss the usefulness of this
approach in statistics for the analysis of deformable models in signal and
image processing. In this setting, the problem of estimating a population
barycenter from n independent and identically distributed random probability
measures is also considered.
