The Generalized Excursion Coupling as the Limit of Concave Optimal Transport

Abstract

The Monge-Kantorovich problem with the Euclidean distance cost is degenerate, typically admitting infinitely many optimal plans. A unique optimal plan is selected by perturbing the distance to a strictly convex or increasing strictly concave cost and passing to the limit: the convex side gives the plan induced by the map which is monotone on each transport ray, while the concave side was understood only on the real line, where Juillet proved that power-cost optimizers converge to the so-called excursion coupling, which is induced by a map when the source is atomless. This paper extends the concave selection to ${\mathbb{R}}^n$ for mutually singular finite positive Borel measures $\mu$ and $\nu$ with equal total mass and finite first moments, assuming that $\mu$ is absolutely continuous with respect to the Lebesgue measure: for a broad class of increasing strictly concave perturbations of the distance with a well-defined first-order profile, the corresponding optimal maps converge in $\mu$-measure to the same intrinsic limit $t_{\#}$, independent of the perturbation family and its profile. The map $t_{\#}$ is obtained by disintegrating the transport along its maximal rays and applying the one-dimensional excursion coupling on each ray. The same limit is established for the power costs $\Vert x-y{\Vert }^{1-\epsilon }$ under a finite $\Vert x\Vert \ln (1+\Vert x\Vert )$ moment condition, establishing a conjecture of Juillet.