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

Statement of Theorem

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

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