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, \mu(0)=\mu, \mu(1)=\nu \} </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, \mu(1)=\nu \} </math> | ||
== Generalization == | |||
= References = | = References = | ||
Revision as of 12:31, 8 June 2020
Introduction
There are many ways that we can characterize Wasserstein metric ... [1]
The main idea is to characterize absolutely (AC) curves ...
Statement of Theorem
- Theorem.(Benamow-Brenier)[1] Let ,
