A view on optimal transport from noncommutative geometry (Q543954)

In this paper the authors discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of the Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry. They prove that on any complete Riemannian spin manifold the two distances coincide. Further, they give some estimates of natural upper and lower bounds to the distance between any two probability distributions on convex manifolds in the sense of the Nash embedding. For the Euclidean space \(\mathbb R^n\), they explicitly compute the distance for a particular class of distributions generalizing the Gaussian wave packet. Focusing on the standard model and the Moyal plane, they explore the analogy between the spectral and the Wasserstein distances in the noncommutative case. In particular they point out that in the two-sheeted space of the standard model, an optimal-transport interpretation of the metric requires a cost function that does not vanish on the diagonal. The later is similar to the cost function occurring in the relativistic heat equation.