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
($\partial _{t}\mu +\nabla (v\mu )=0$).

Geodesics in general metric spaces

First, we will introduce definition of the geodesic in general metric space $X$. We are going to follow presentation from the book by Santambrogio.

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

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

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 is geodesic curve. This is very important definition, since for $p>1$ we have that every constant-speed geodesic $\omega$ is also in $AC(X)$

Statement of Theorem

Now, we can rephrase Wasserstein metrics in dynamic language. In special case, for $p=2$:

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

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

Generalization

It is possible to generalize the previous theorem and theory to $W_{p}$ metrics. More about that could be seen in the book .

However, it is possible to generalize theorem for a different kind of geodesics .