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 $L(\omega )$ of 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. More about it 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.

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.

Statement of Theorem

Now, we can rephrase Wasserstein metrics in dynamic language. In special case:

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

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

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 .