  Consider a multi-agent system whereby each agent has an initial probability
measure. In this paper, we propose a distributed algorithm based upon
stochastic, asynchronous and pairwise exchange of information and displacement
interpolation in the Wasserstein space. We characterize the evolution of this
algorithm and prove it computes the Wasserstein barycenter of the initial
measures under various conditions. One version of the algorithm computes a
standard Wasserstein barycenter, i.e., a barycenter based upon equal weights;
and the other version computes a randomized Wasserstein barycenter, i.e., a
barycenter based upon random weights for the initial measures. Finally, we
specialize our results to Gaussian distributions and draw a connection with the
modeling of opinion dynamics in mathematical sociology.
