发表机构
Columbia University（哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明在欧氏空间中，对距离进行严格凹扰动后取极限，最优输运映射收敛到与扰动无关的广义游走耦合，该耦合通过沿最大射线分解并应用一维游走耦合构造，并证实了Juillet关于幂代价的猜想。
AI 中文摘要
具有欧几里得距离代价的Monge-Kantorovich问题是退化的，通常允许无穷多个最优计划。通过将距离扰动为严格凸或严格递增凹代价并取极限，可选出唯一的最优计划：凸侧给出由在每个输运射线上单调的映射诱导的计划，而凹侧仅在实直线上被理解，其中Juillet证明了幂代价优化子收敛到所谓的游走耦合，当源无原子时该耦合由映射诱导。本文将凹选择扩展到$\mathbb{R}^n$，针对具有相等总质量和有限一阶矩的互奇异有限正Borel测度$\mu$和$\nu$，假设$\mu$关于Lebesgue测度绝对连续：对于一大类具有良定义一阶轮廓的距离的严格递增凹扰动，相应的最优映射在$\mu$-测度下收敛到相同的固有极限$t_\\#$，该极限独立于扰动族及其轮廓。映射$t_\\#$通过沿其最大射线分解输运并在每条射线上应用一维游走耦合获得。在有限$\\|x\\|\ln(1+\\|x\\|)$矩条件下，对幂代价$\\|x-y\\|^{1-\varepsilon}$也建立了相同的极限，从而证实了Juillet的一个猜想。
英文摘要
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 $μ$ and $ν$ with equal total mass and finite first moments, assuming that $μ$ 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 $μ$-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 $\|x-y\|^{1-\varepsilon}$ under a finite $\|x\|\ln(1+\|x\|)$ moment condition, establishing a conjecture of Juillet.
