# On a solution to the Monge transport problem on the real line arising from the strictly concave case

## TL;DR

This paper introduces a novel approach called excursion coupling to ensure uniqueness in the Monge transport problem on the real line for costs that are powers less than one, by analyzing limits as the power approaches one.

## Contribution

It provides a complete construction and characterization of the excursion coupling, a new solution concept for the Monge problem with strictly concave costs on the real line.

## Key findings

- Unique solution constructed via excursion coupling
- Characterization of transport routes through combinatoric and geometric methods
- Solution as a limit of transport problems with power costs less than one

## Abstract

It is well-known that the optimal transport problem on the real line for the classical distance cost may not have a unique solution. In this paper we recover uniqueness by considering the transport problems where the costs are a power smaller than one of the distance, and letting this parameter tend to one. A complete construction of this solution that we call excursion coupling is given. This is reminiscent to the one in the convex case. It is also characterized as the solution of secondary transport problems. Moreover, a combinatoric/geometric characterization of the routes used for this transport plan is provided.