  Distributed consensus in the Wasserstein metric space of probability measures
on the real line is introduced in this work. Convergence of each agent's
measure to a common measure is proven under a weak network connectivity
condition. The common measure reached at each agent is one minimizing a
weighted sum of its Wasserstein distance to all initial agent measures. This
measure is known as the Wasserstein barycenter. Special cases involving
Gaussian measures, empirical measures, and time-invariant network topologies
are considered, where convergence rates and average-consensus results are
given. This work has possible applicability in computer vision, machine
learning, clustering, and estimation.
