Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Optimal transport between random measures

Martin Huesmann

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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.


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

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

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

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

Optimal transport Monge solution Random measure Equivariance Amenable Allocation Matching Tessellation


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

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