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 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 three assumptions on a given dN -monotone transference plan π which imply respectively: strongly 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 π. We apply these results to the Monge problem in Rd 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.