Transport Problems and Disintegration Maps

TL;DR

This paper introduces transport classes via disintegration of plans, framing the Monge problem as a special case of Kantorovich transport within fixed classes, and explores the solvability of these constrained problems.

Contribution

It formalizes the concept of transport classes and links the Monge problem to Kantorovich transport, providing a new perspective on constrained transport problems.

Findings

Monge problem is a special case within fixed transport classes.

Transport problems constrained to a class are equivalent to Monge problems over Wasserstein spaces.

Solvability of constrained problems varies, with Monge being a fortunate case.

Abstract

By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem constrained to a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures. Concerning solvability of this kind of constrained problems, it turns out that in some sense the Monge problem corresponds to a lucky case.