  In this paper, we focus on the analysis of the regularized Wasserstein
barycenter problem. We provide uniqueness and a characterization of the
barycenter for two important classes of probability measures: (i) Gaussian
distributions and (ii) $q$-Gaussian distributions; each regularized by a
particular entropy functional. We propose an algorithm based on gradient
projection method in the space of matrices in order to compute these
regularized barycenters. We also consider a general class of
$\varphi$-exponential measures, for which only the non-regularized barycenter
is studied. Finally, we numerically show the influence of parameters and
stability of the algorithm under small perturbation of data.
