Invariance properties of the Monge-Kantorovich mass transport problem

Understand

We consider the multidimensional Monge-Kantrovich 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 Kantrovich 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.