Geodesics and generalized geodesics

$W_{2}(\mu ,\nu ):=\min _{\gamma \in \Gamma (\mu ,\nu )}\left(\int |x_{1}-x_{2}|^{2}\,d\gamma (x_{1},x_{2})\right)^{1/2}$

Statement of Theorem

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

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