Optimal transport from Lebesgue to Poisson

By Martin Huesmann, Karl‐Theodor Sturm. 2010

48 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/50b775509c07e16b97a06efe7ef12d37305a3e0b

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In Applied Mathematics

In Statistics and Probability

Citations in the corpus

Constructive quantization: Approximation by empirical measures (2011, background): “Similar problems arise in [13] for optimal transports from Poisson point processes with Lebesgue intensity to Lebesgue measure.”
A PDE approach to a 2-dimensional matching problem (2016, background): “optimal assignment problem, [C04], where the optimization involves also the weights of the Dirac masses δXi and the closely related problem of transporting Lebesgue measure to a Poisson point process [HS13], which involves in the limit measures with inﬁnite mass. In this paper we focus on two of the”