Optimal transport from Lebesgue to Poisson

Huesmann, Martin; Sturm, Karl-Theodor

doi:10.1214/12-aop814

Cited by 35 publications

(41 citation statements)

References 31 publications

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“…As a special case of our result, we recover the results by Huesmann and Sturm in [HS10]. They studied couplings between the Lebesgue measure and an equivariant point process of intensity β ∈ (0, 1].…”

Section: Introduction and Statement Of Main Resultssupporting

confidence: 82%

“…They showed that there is a unique optimal semicoupling and also proved an approximation result by solutions to transport problems on bounded regions. Furthermore, in [HS10] necessary and sufficient conditions have been derived implying the finiteness of the mean asymptotic transportation cost in the case of transporting the Lebesgue measure to a Poisson point process. By applying the same techniques similar estimates can be achieved for the case of a compound Poisson process with iid weights (X i ) i∈N , i.e.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

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Optimal transport between random measures

Huesmann¹

2016

Ann. Inst. H. Poincaré Probab. Statist.

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We study couplings q • of two equivariant random measures λ • and µ • on a Riemannian manifold (M, d, m). Given a cost function we ask for minimizers of the mean transportation cost per volume. In case the minimal/optimal cost is finite and λ ω m we prove that there is a unique equivariant coupling minimizing the mean transportation cost per volume. Moreover, the optimal coupling is induced by a transportation map, q • = (id, T ) * λ • . We show that the optimal transportation map can be approximated by solutions to classical optimal transportation problems on bounded regions. In case of L p −cost the optimal transportation cost per volume defines a metric on the space of equivariant random measure with unit intensity.

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Section: Introduction and Statement Of Main Resultssupporting

confidence: 82%

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Optimal transport between random measures

Huesmann¹

2016

Ann. Inst. H. Poincaré Probab. Statist.

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“…Let us close this note by pointing out that in dimension d ≥ 3, the optimal transport plans corresponding to a very closely related optimal matching problem, have been used in [21] to construct in the limit L → ∞, a stationary and locally optimal coupling between the Poisson point process on R d and the Lebesgue measure. For d = 2, such a coupling is expected not to exist.…”

Section: Application To the Optimal Matching Problemmentioning

confidence: 99%

A variational approach to regularity theory in optimal transportation

Goldman¹

2019

Séminaire Laurent Schwartz — EDP Et Applications

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This paper describes recent results obtained in collaboration with M. Huesmann and F. Otto on the regularity of optimal transport maps. The main result is a quantitative version of the well-known fact that the linearization of the Monge-Ampère equation around the identity is the Poisson equation. We present two applications of this result. The first one is a variational proof of the partial regularity theorem of Figalli and Kim and the second is the rigorous validation of some predictions made by Carraciolo and al. on the structure of the optimal transport maps in matching problems.

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“…The landmark in this topic is [3] which generalizes [10] to arbitrary dimensions. There are several other constructions in the literature for the case of simple point processes, such as gravitational allocation [1], optimal transport [6], one-sided stable allocation on the line [9], etc.…”

Section: Stable Transports Between Stationary Random Measuresmentioning

confidence: 99%

Stable transports between stationary random measures

Haji-Mirsadeghi¹,

Khezeli²

2016

Electron. J. Probab.

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We give an algorithm to construct a translation-invariant transport kernel between ergodic stationary random measures $\Phi$ and $\Psi$ on $\mathbb R^d$, given that they have equal intensities. As a result, this yields a construction of a shift-coupling of an ergodic stationary random measure and its Palm version. This algorithm constructs the transport kernel in a deterministic manner given realizations $\varphi$ and $\psi$ of the measures. The (non-constructive) existence of such a transport kernel was proved in [8]. Our algorithm is a generalization of the work of [3], in which a construction is provided for the Lebesgue measure and an ergodic simple point process. In the general case, we limit ourselves to what we call constrained densities and transport kernels. We give a definition of stability of constrained densities and introduce our construction algorithm inspired by the Gale-Shapley stable marriage algorithm. For stable constrained densities, we study existence, uniqueness, monotonicity w.r.t. the measures and boundedness.Comment: In the second version, we change the way of presentation of the main results in Section 4. The main results and their proofs are not changed significantly. We add Section 3 and Subsection 4.6. 25 pages and 2 figure

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Optimal transport from Lebesgue to Poisson

Cited by 35 publications

References 31 publications

Optimal transport between random measures

Optimal transport between random measures

A variational approach to regularity theory in optimal transportation

Stable transports between stationary random measures

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