We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dLis a geodesic Borel distance which makes (X, dL) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem π which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1-dimensional Hausdorff distance induced by dL. It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting dL-cyclical monotonicity is not sufficient for optimality.

1 aBianchini, Stefano1 aCavalletti, Fabio uhttps://doi.org/10.1007/s00220-013-1663-800591nas a2200145 4500008004100000022001400041245003800055210003400093260000800127300001400135490000700149520022100156100002200377856004600399 2012 eng d a1432-083500aThe Monge problem in Wiener space0 aMonge problem in Wiener space cSep a101–1240 v453 aWe address the Monge problem in the abstract Wiener space and we give an existence result provided both marginal measures are absolutely continuous with respect to the infinite dimensional Gaussian measure γ.

1 aCavalletti, Fabio uhttps://doi.org/10.1007/s00526-011-0452-500376nas a2200109 4500008004100000245007700041210006900118300001200187490000700199100002200206856003800228 2012 eng d00aOptimal Transport with Branching Distance Costs and the Obstacle Problem0 aOptimal Transport with Branching Distance Costs and the Obstacle a454-4820 v441 aCavalletti, Fabio uhttps://doi.org/10.1137/10080143301064nas a2200193 4500008004100000020002200041245004100063210003700104260002800141300001400169520049000183100002300673700002200696700002100718700002400739700001900763700001600782856007200798 2011 eng d a978-1-4419-9554-400aThe Monge Problem in Geodesic Spaces0 aMonge Problem in Geodesic Spaces aBoston, MAbSpringer US a217–2333 aWe address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish non branching geodesic space. We show that we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce an assumption on the transport problem π which implies that the conditional probabilities of the first marginal on each geodesic are continuous. It is known that this regularity is sufficient for the construction of an optimal transport map.

1 aBianchini, Stefano1 aCavalletti, Fabio1 aBressan, Alberto1 aChen, Gui-Qiang, G.1 aLewicka, Marta1 aWang, Dehua uhttps://www.math.sissa.it/publication/monge-problem-geodesic-spaces