Geodesics and generalized geodesics
There are many ways that we can describe Wasserstein metric. One of them is to characterize absolutely continuos curves (AC)(p.188) and provide a dynamic formulation of the special case Namely, it is possible to see as an infimum of the lengts of curves that satisfy Continuity equation.
Geodesics in general metric spaces
First, we will introduce definition of the geodesic in general metric space . In the following sections. we are going to follow a presentation from the book by Santambrogio with some digression, here and there.
For the starting point, we need to introduce length of the curve in our metric space .
- Definition. A length of the curve is defined by
Secondly, we use the definition of length of a curve to introduce a geodesic curve.
- Definition. A curve is said to be geodesic between and in if it minimizes the length among all the curves
such that and .
Since we have a definition of a geodesic in the general metric space, it is natural to think of Riemannian structure. It can be formally defined. More about this topic can be seen in the following article Formal Riemannian Structure of the Wasserstein_metric.
Now, we proceed with necessary definitions in order to be able to understand Wasserstein metric in a different way.
- Definition. A metric space is called a length space if it holds
A space is called geodesic space if the distance is attained for some curve .
- Definition. In a length space, a curve is said to be constant speed geodesic between and in if it satisfies
It is clear that constant-speed geodesic curve connecting and is a geodesic curve. This is very important definition since we have that every constant-speed geodesic is also in where almost everywhere in .
In addition, minimum of the set is attained by our constant-speed geodesic curve Last fact is important since it is connected to Wasserstein metric. For more information, please take a look at Wasserstein metric.
For more information on constant-speed geodesics, especially how they depend on uniqueness of the plan that is induced by transport and characterization of a constant-speed geodesic look at the book by L.Ambrosio, N.Gilgi, G.Savaré  or the book by Santambrogio.
Dynamic formulation of Wasserstein distance
Finally, we can rephrase Wasserstein metrics in dynamic language as mentioned in the Introduction.
Whenever is convex set, is a geodesic space. Proof can be found in the book by Santambrogio.
- Theorem. Let . Then
In special case, when is compact, infimum is attained by some constant-speed geodesic.
There are many ways to generalize this fact. We will talk about a special case and a displacement convexity. Here we follow again book by Santambrogio.
In general, the functional is not a displacement convex. We can fix this by introducing a generalized geodesic.
- Definition. Let be an absolutely continuous measure and and probability measures in . We say that
is a generalized geodesic in with base , where is the optimal transport plan from to and is the optimal transport plan from to .
By calculation, we have the following
Therefore, along the generalized geodesic, the functional is convex.
This fact is very important in establishing uniqueness and existence theorems in the geodesic flows.
- F. Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 1, pages 202-207
- [https://link.springer.com/book/10.1007/b137080 L.Ambrosio, N.Gilgi, G.Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Chapter 7.2., pages 158-160]
- F. Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 1, pages 269-276