Transport Plan, Transport Map and Transport Path—Difference of the Optimal Transport Problem between ℝ² and Finite Set—

Abstract

This paper is concerned with a variational problem on the space of probability measures over either ℝ² or a finite set, the so-called optimal transport problem. We discussed three ways to formulate the optimal transport problem on ℝ² as follows: the use of a probability measure (transport plan) on ℝ² × ℝ², a map (transport map) from ℝ² to ℝ², and a family of paths (transport path) on ℝ². The definition of transport path requires the property that each pair of points in ℝ² should be connected by a length minimizing curve. To define transport paths on a finite set, we explained how to modify the optimal transport problem on a finite set.