Optimal transportation of measures with a parameter

TL;DR

This paper studies how optimal transportation plans and costs vary continuously with parameters in metric spaces, providing conditions for continuity, existence, and convergence of optimal plans and mappings.

Contribution

It establishes the continuity of optimal transportation costs and plans with respect to parameters, and provides new results on convergence of Monge mappings under parameter variations.

Findings

Continuity of transportation cost with respect to parameters.

Existence of continuous approximate optimal plans.

Counterexamples for continuous selection of optimal plans.

Abstract

We consider optimal transportation of measures on metric and topological spaces in the case where the cost function and marginal distributions depend on a parameter with values in a metric space. The Hausdorff distance between the sets of probability measures with given marginals is estimated via the distances between the marginals themselves. The continuity of the cost of optimal transportation with respect to the parameter is proved in the case of continuous dependence of the cost function and marginal distributions on this parameter. Existence of approximate optimal plans continuous with respect to the parameter is established. It is shown that the optimal plan is continuous with respect to the parameter in case of uniqueness. Examples are constructed when there is no continuous selection of optimal plans. Finally, a general result on convergence of Monge optimal mappings is proved.