Optimal Transport with Branching Distance Costs and the Obstacle Problem

We address the Monge problem in metric spaces with a geodesic distance: $(X,d)$ is a Polish space and $d_N$ is a geodesic Borel distance which makes $(X,d_N)$ 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 a family of geodesics. We introduce three assumptions on a given $d_{N}$-monotone transference plan $\pi$ which imply, respectively, strong consistency of disintegration, continuity of the conditional probabilities of the first marginal, and a regularity property for the geometry of chain of transport rays. We show that this regularity is sufficient for the construction of a transport map with the same transport cost of $\pi$. We apply these results to the Monge problem in $\mathbb{R}^{d}$ with smooth, convex, and compact obstacle obtaining the existence of an optimal map, provided the first marginal is absolutely continuous with respect to the d-dimensional Lebesgue measure.