Geodesics and generalized geodesics: Difference between revisions
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: '''Theorem.'''(Benamow-Brenier)<ref name=Santambrogio /> Let <math> \mu, \nu \in P_{2}(R^{d}) </math>, <br> | : '''Theorem.'''(Benamow-Brenier)<ref name=Santambrogio /> Let <math> \mu, \nu \in P_{2}(R^{d}) </math>, <br> | ||
<math> 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 </math> | <math> 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 </math> | ||
= References = | = References = | ||
Revision as of 11:39, 8 June 2020
Statement of Theorem
- Theorem.(Benamow-Brenier)[1] Let ,
Failed to parse (unknown function "\vu"): {\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 }