UC Davis

Optimal transportation studies the transportation of a given mass distribution to a designatedmass distribution so that a given transport cost function reaches minimum. Under different formulations of transport problems, including the Monge transport problem, the Kantorovich transport problem, and the ramified transport problem, transportation has various characterizations. In this dissertation, we use transport paths from ramified transportation to characterize transportation, and use measures and currents to characterize transport paths. We show that a good decomposition of a transport path can be refined into a better decomposition that is more cycle “sensitive”. Using better decomposition, we show that cycle-free transport paths can be decomposed into map-compatible transport paths components, and we also prove similar results when transport paths are under capacity constraint. These decomposition results describe properties of optimal transport paths, and using these properties we can narrow down the scope of finding an optimal transport path. The notion of capacity constraint on transport paths is a generalization of the usual transport paths, and it makes transport paths more relevant and applicable to real life transportation. In Chapter 1, we first review concepts related to measures, then introducing the Monge and the Kantorovich transport problems. In the Monge and the Kantorovich transport problem, transportation is characterized by functions defined on sources and targets, rather than the actual transport path connecting them. In the next chapter, we will see another characterization of transportation using transport paths. In Chapter 2, we introduce ramified transportation, which uses the actual transport paths from sources to targets to characterize the transportation between two mass distributions. Transport paths in ramified transportation can be described using rectifiable 1-currents, and this is where we start in this chapter. In Chapter 3, we first revise good decompositions of a transport path into better decompositions which are used later to decompose cycle-free transport paths based on targets. Then we show the components of previously decomposed transport paths are compatible with certain transport maps and plans. Finally we consider a special type of transport paths, the stair-shaped transport paths, which can be decomposed as the difference of two map-compatible transport paths. In Chapter 4, we study transport paths under capacity constraint. In this case, each transport path is defined through multiple components such that the total mass transported on each transport path component is no more than the predetermined capacity. Then we start to analyze the amount of components needed in a transportation, the existence of admissible optimal transport paths, regularities among different transport path components, and existence of map-compatibility.