Geodesics and generalized geodesics

Introduction

There are many ways that we can characterize Wasserstein metric ...

The main idea is to characterize absolutely (AC) curves ...

Statement of Theorem

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

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