Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Optimal transport between random measures

Martin Huesmann

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Abstract

We analyze the optimal transport problem between two equivariant random measures and derive sufficient conditions for the existence of a unique Monge solution. Moreover, we show that equivariance naturally appears in this context by proving that classical optimal couplings on bounded sets converge to the optimal coupling on the whole space. Finally, we derive sufficient conditions for the $L^{p}$ cost to be finite by introducing a suitable metric.

Résumé

Nous analysons le problème du transport optimal entre deux mesures aléatoires et équivariantes et démontrons des conditions qui garantissent l’existence d’une solution de type Monge. En outre, nous démontrons que l’équivariance apparaît naturellement dans ce contexte en prouvant que les couplages optimaux classiques dans des ensembles bornés convergent vers le couplage optimal dans tout l’espace. Finalement nous démontrons des conditions suffisantes pour que le coût au sens $L^{p}$ soit fini en introduisant une métrique appropriée.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 1 (2016), 196-232.

Dates
Received: 13 October 2013
Revised: 17 June 2014
Accepted: 15 July 2014
First available in Project Euclid: 6 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1452089267

Digital Object Identifier
doi:10.1214/14-AIHP634

Mathematical Reviews number (MathSciNet)
MR3449301

Zentralblatt MATH identifier
1187.60030

Subjects
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 43A07: Means on groups, semigroups, etc.; amenable groups

Keywords
Optimal transport Monge solution Random measure Equivariance Amenable Allocation Matching Tessellation

Citation

Huesmann, Martin. Optimal transport between random measures. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 1, 196--232. doi:10.1214/14-AIHP634. https://projecteuclid.org/euclid.aihp/1452089267


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