2023/07/17 by Mariem Abdellatif, Abdellatif, Mariem, Peter Kuchling +5
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2307.08402
openalex publication_date 2023/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we represent the Wasserstein metric of order p, where p∈ [1,∞), in terms of the comonotonicity copula, for the case of probability measures on \Rd, by revisiting existing results. In 1973, Vallender established the link between the 1-Wasserstein metric and the corresponding distribution functions for d=1. In 1956 Giorgio dall'Aglio showed that the p-Wasserstein metric in d=1 could be written in terms of the comonotonicity copula M without being aware of the concept of copulas or Wasserstein metrics. In this article, for the proofs we explicitly combine tools from copula theory and Wasserstein metrics. The extension to general d∈\N has some restriction, as discussed e.g. in \citeAlfonsi and \citeBDS. Some of the results of \citeAlfonsi, \citeBDS and \citeRR are revisited here in a more explicit form in terms of the comonotonicity copula.