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For the basic case of $L_2$ optimal transport between two probabilitymeasures on a Euclidean space, the regularity of the coupling measure and thetransport map in the tail regions of these measures is studied. For thispurpose, Robert McCann's classical existence and uniqueness results areextended to a class of possibly infinite measures, finite outsideneighbourhoods of the origin. For convergent sequences of pairs of suchmeasures, the stability of the multivalued transport maps is considered, and auseful notion of locally uniform convergence of these maps is verified underlight assumptions. Applied to regularly varying probability measures, thesegeneral results imply the existence of tail limits of the transport plan andthe coupling measure, these objects exhibiting distinct types of homogeneity.

Tails of optimal transport plans for regularly varying probability measures