OPTIMAL TRANSPORT FROM LEBESGUE TO POISSON

• 成果类型: Article
• 署名作者: Huesmann, Martin; Sturm, Karl-Theodor
• 署名单位: University of Bonn
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP814
• 发表日期: 2013
• 页码: 2426-2478
• 关键词: metric-measure-spaces; geometry; EQUATIONS

摘要

This paper is devoted to the study of couplings of the Lebesgue measure and the Poisson point process. We prove existence and uniqueness of an optimal coupling whenever the asymptotic mean transportation cost is finite. Moreover, we give precise conditions for the latter which demonstrate a sharp threshold at d = 2. The cost will be defined in terms of an arbitrary increasing function of the distance. The coupling will be realized by means of a transport map (allocation map) which assigns to each Poisson point a set (cell) of Lebesgue measure 1. In the case of quadratic costs, all these cells will be convex polytopes.