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 $\mu ,\nu \in P_{2}(R^{d})$,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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, \quad \mu(1)=\vu }