\(W_{1,+}\)-interpolation of probability measures on graphs. (English) Zbl 1347.49077

Hillion, Erwan

Electron. J. Probab. 19, Paper No. 92, 29 p. (2014).

Summary: We generalize an equation introduced by Benamou and Brenier and 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 distribution, \((f_0(x))_{x \in G}\) and \((f_1(x))_{x \in G}\), we prove the existence of a curve \((f_t(x))_{t\in [0,1], x\in G}\) 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 a solution of a certain optimization problem.

MSC:

49Q20	Variational problems in a geometric measure-theoretic setting
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)
60D05	Geometric probability and stochastic geometry
60B10	Convergence of probability measures

Keywords:

optimal transport; Wasserstein distances; discrete probability