# American Institute of Mathematical Sciences

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May  2016, 36(5): 2653-2671. doi: 10.3934/dcds.2016.36.2653

## Invariance properties of the Monge-Kantorovich mass transport problem

 1 School of Mathematics and Statistics, Carleton University, Ottawa, K1S5B6, Canada

Received  April 2015 Revised  July 2015 Published  October 2015

We consider the multimarginal Monge-Kantorovich transport problem in an abstract setting. Our main results state that if a cost function and marginal measures are invariant by a family of transformations, then a solution of the Kantorovich relaxation problem and a solution of its dual can be chosen so that they are invariant under the same family of transformations. This provides a new tool to study and analyze the support of optimal transport plans and consequently to scrutinize the Monge problem. Birkhoff's Ergodic theorem is an essential tool in our analysis.
Citation: Abbas Moameni. Invariance properties of the Monge-Kantorovich mass transport problem. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2653-2671. doi: 10.3934/dcds.2016.36.2653
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