$W_{1,+}$-interpolation of probability measures on graphs

We generalize an equation introduced by Benamou and Brenier, characterizing Wasserstein W_p-geodesics for p > 1, from the continuous setting of probability distributions on a Riemannian manifold to the discrete setting of probability distributions on a general graph. Given an initial and a final distributions f_0 and f_1, we prove the existence of a curve (f_t) satisfying this Benamou-Brenier equation. We also show that such a curve can be described as a mixture of binomial distributions with respect to a coupling that is solution of a certain optimization problem.