Excursion Coupling for the Monge Problem on the Line (Juillet 2019)Research Paper
The Monge optimal transport problem on the real line with the classical distance cost $|x-y|$ famously fails to have a unique solution. Juillet (2019) restored uniqueness by considering the strictly concave power costs $|x-y|^p$ with $p<1$ and letting $p\to 1^-$: the limit selects a distinguished optimal plan, the *excursion coupling*, built from the level sets of the difference $F_\sigma=F_\mu-F_\nu$ of the cumulative distribution functions. This mission formalizes the completed-graph construction, the generalized Banach indicatrix identities of Bertoin-Yor, the alternating crossing structure of almost every level, and the marginal identities for the crossing counting measures. It culminates in Propositions 3.5-3.6: every monotone transport plan is concentrated on the paired routes, and the marginals uniquely determine the coupling carried by those routes, including in the presence of atoms. This mission formalizes the key implication 3=>4 in Juillet's Main Theorem.