Optimal transport with branching distance costs and the obstacle problem

TL;DR

This paper studies the Monge optimal transport problem in geodesic metric spaces with branching distances, establishing conditions for the existence of optimal transport maps even in the presence of obstacles.

Contribution

It introduces new assumptions on transference plans that enable reduction to geodesic transport problems and proves the existence of optimal maps with obstacles.

Findings

Reduction of the Monge problem to geodesic transport problems.

Existence of optimal transport maps under new regularity assumptions.

Application to obstacle problems with smooth, convex, compact obstacles.

Abstract

We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dN is a geodesic Borel distance which makes (X,dN) a possibly branching geodesic space. We show that under some assumptions on the transference plan we can reduce the transport problem to transport problems along family of geodesics. We introduce two assumptions on the transference plan {\pi} which imply that the conditional probabilities of the first marginal on each family of geodesics are continuous and that each family of geodesics is a hourglass-like set. We show that this regularity is sufficient for the construction of a transport map. We apply these results to the Monge problem in d with smooth, convex and compact obstacle obtaining the existence of an optimal map provided the first marginal is absolutely continuous w.r.t. the d-dimensional Lebesgue measure.