The Annals of Probability
- Ann. Probab.
- Volume 41, Number 4 (2013), 2426-2478.
Optimal transport from Lebesgue to Poisson
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.
Ann. Probab., Volume 41, Number 4 (2013), 2426-2478.
First available in Project Euclid: 3 July 2013
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 49Q20: Variational problems in a geometric measure-theoretic setting
Huesmann, Martin; Sturm, Karl-Theodor. Optimal transport from Lebesgue to Poisson. Ann. Probab. 41 (2013), no. 4, 2426--2478. doi:10.1214/12-AOP814. https://projecteuclid.org/euclid.aop/1372859757