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.
