Geodesics and generalized geodesics

Introduction

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 $W_{2}^{2}$ Namely, it is possible to see $W_{2}^{2}(\mu ,\nu )$ 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 $X$. In this and next section. we are going to follow a presentation from the book by Santambrogio with some digression, here and there.

Definition. A curve $c:[0,1]\rightarrow X$ is said to be geodesic in $X$ if it minimizes the length $L(\omega )$ among all the curves $\omega :[0,1]\rightarrow X$
such that $c(0)=\omega (0)$ and $c(1)=\omega (1)$.

Since we have a definition of a geodesic in the general space, it is natural to think of Riemannian structure. It can be 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 $(X,d)$ is called a length space if it holds

                    $d(x,y)=\inf\{L(\omega )|\quad \omega \in AC(X),\quad \omega (0)=x\quad \omega (1)=y\}.$

A space $(X,d)$ is called geodesic space if the distance $d(x,y)$ is attained for some curve $\omega$.

Definition. In a length space, a curve $\omega :[0,1]\rightarrow X$ is said to be constant speed geodesic between $\omega (0)$ and $\omega (1)$ in $X$ if it satisfies

                    $d(\omega (s),\omega (t))=|t-s|d(\omega (0),\omega (1))$ for all $t,s\in [0,1]$

It is clear that constant-speed geodesic curve $\omega$ connecting $x$ and $y$ is a geodesic curve. This is very important definition since,for $p>1$, we have that every constant-speed geodesic $\omega$ is also in $AC(X)$ where $|\omega '(t)|=d(\omega (0),\omega (1))$ almost everywhere in $[0,1]$.
In addition, minimum of the set $\{\int _{0}^{1}|c'(t)|^{p}dt|c:[0,1]\rightarrow X,c(0)=x,c(1)=y\}$ is attained by our constant-speed geodesic curve $\omega .$

For more information on constant-speed geodesics, especially how they depend on uniqueness of the plan that is induced by transport look at the book "Gradient Flows in Metric Spaces and in the Space of Probability Measures" by L.Ambrosio, N.Gilgi, G.Savaré .

Statement of Theorem

Finally, we can rephrase Wasserstein metrics in dynamic language as mentioned in the Introduction.

Whenever $\Omega \subseteq {\mathcal {R}}^{d}$ is convex set and $p>1$, $W_{p}(\Omega )$ is a geodesic space. Proof can be found in the book by Santambrogio.

Theorem.(Benamou-Brenier) Let $\mu ,\nu \in {\mathcal {P}}_{2}(R^{d})$. Then we have

      $W_{p}^{p}(\mu ,\nu )=\inf _{(\mu (t).\nu (t))}\{\int _{0}^{1}|v(,t)|_{L^{2}(\mu (t))}^{2}dt\quad |\quad \partial _{t}\mu +\nabla (v\mu )=0,\quad \mu (0)=\mu ,\quad \mu (1)=\nu \}.$

In special case, when $\Omega$ is compact, infimum is attained by some constant-speed geodesic.