  In this paper, a regularization of Wasserstein barycenters for random
measures supported on $\mathbb{R}^{d}$ is introduced via convex penalization.
The existence and uniqueness of such barycenters is first proved for a large
class of penalization functions. The Bregman divergence associated to the
penalization term is then considered to obtain a stability result on penalized
barycenters. This allows the comparison of data made of $n$ absolutely
continuous probability measures, within the more realistic setting where one
only has access to a dataset of random variables sampled from unknown
distributions. The convergence of the penalized empirical barycenter of a set
of $n$ iid random probability measures towards its population counterpart is
finally analyzed. This approach is shown to be appropriate for the statistical
analysis of either discrete or absolutely continuous random measures. It also
allows to construct, from a set of discrete measures, consistent estimators of
population Wasserstein barycenters that are absolutely continuous.
