The optimal mass transport problem for relativistic costs

Bertrand, Jérôme; Puel, Marjolaine

doi:10.1007/s00526-011-0485-9

Cited by 27 publications

(44 citation statements)

References 29 publications

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“…The conditions are met on a weakly dense subset of pairs of measures by Corollary 2.9. Theorem 2.8 generalizes results in [6,7,17]. The condition of strict timelikeness is related to parts of the definition of q-separated measures in [17].…”

Section: Introductionsupporting

confidence: 61%

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On the existence of dual solutions for Lorentzian cost functions

Kell

Suhr

2020

Ann. Inst. H. Poincaré Anal. Non Linéaire

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The dual problem of optimal transportation in Lorentz-Finsler geometry is studied. It is shown that in general no solution exists even in the presence of an optimal coupling. Under natural assumptions dual solutions are established. It is further shown that the existence of a dual solution implies that the optimal transport is timelike on a set of full measure. In the second part the persistence of absolute continuity along an optimal transportation under obvious assumptions is proven and a solution to the relativistic Monge problem is provided. resultsLet M be a smooth manifold. Throughout the article one fixes a complete Riemannian metric h on M , though local changes to the metric will be allowed. Consider a continuous function L :denotes the zero section in T M ) and positive homogenous of degree 2 such that the second fiber derivative is non-degenerate with index dim M − 1. Let C be a causal structure of (M, L), see [21], and define the Lagrangian L on T M by settingThe pair (M, L) is referred to as a Lorentz-Finsler manifold.One calls an absolutely continuous curve γ : I → M (C-)causal ifγ ∈ C for almost all t ∈ I. Note that this condition already implies that the tangent vector is contained in C whenever it exists.Denote with J + (x) the set of points y ∈ M such that there exists a causal curve γ : [a, b] → M with γ(a) = x and γ(b) = y. Two points x and y will be called causally related if y ∈ J + (x). Note that this relation is in general asymmetric. Define the setand J − (y) : {x ∈ M | y ∈ J + (x)}. A Lorentz-Finsler manifold is said to be causal if it does not admit a closed causal curve, i.e. J + ∩ △ = ∅ for △ := {(x, x)| x ∈ M }. Definition 2.1. A causal Lorentz-Finsler manifold (M, L) is globally hyperbolic if the sets J + (x) ∩ J − (y) are compact for all x, y ∈ M .

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Section: Introductionsupporting

confidence: 61%

“…The theorem generalizes [6,Theorem 5.13] and [7,Theorem B]. Note that Theorem 2.8 is most likely optimal, as Theorem 2.12 shows that dual solutions cannot exist whenever there are optimal couplings transporting a set of positive measure along ∂C, the boundary of the lightcone.…”

Section: Proposition 23 ([21]mentioning

confidence: 80%

On the existence of dual solutions for Lorentzian cost functions

Kell

Suhr

2020

Ann. Inst. H. Poincaré Anal. Non Linéaire

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“…In this paper, we consider the classical mass transport problem with a particular class of cost functions c. While the case when c is strictly convex and real-valued is well understood (we refer to [34] for a survey), our goal is to continue the study of a wide class of non real-valued cost functions, called "relativistic costs", which were introduced in [12] as a generalization to the "relativistic heat cost".…”

Section: Introductionmentioning

confidence: 99%

“…Afterwards, the corresponding relativistic heat equation has also been studied by McCann and Puel in [28] by means of the Jordan-Kinderlehrer-Otto approach introduced in [25], and by Andreu, Caselles and Mazón via PDE methods in a series of papers [3,5,4,6,7,8,9,16]. We refer to [14,12] and the references therein for more on this topic. Notice that, from the mathematical point of view, a quite interesting feature of this cost function is that it is strictly convex and bounded on its domain, hence in particular it is not continuous on R n × R n (while the non real-valued cost functions have been often considered as continuous, see for instance [23,22,29,30,33,11]).…”

Section: Introductionmentioning

confidence: 99%

Kantorovich potentials and continuity of total cost for relativistic cost functions

Bertrand

Pratelli

Puel³

2018

Journal de Mathématiques Pures et Appliquées

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In this paper we consider the optimal mass transport problem for relativistic cost functions, introduced in [12] as a generalization of the relativistic heat cost. A typical example of such a cost function is ct(x, y) = h(y−x t), h being a strictly convex function when the variable lies on a given ball, and infinite otherwise. It has been already proved that, for every t larger than some critical time T > 0, existence and uniqueness of optimal maps hold; nonetheless, the existence of a Kantorovich potential is known only under quite restrictive assumptions. Moreover, the total cost corresponding to time t has been only proved to be a decreasing rightcontinuous function of t. In this paper, we extend the existence of Kantorovich potentials to a much broader setting, and we show that the total cost is a continuous function. To obtain both results the two main crucial steps are a refined "chain lemma" and the result that, for t > T , the points moving at maximal distance are negligible for the optimal plan.

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“…The latter is a new and fast-developing area of research [10,13,55], 1 which has found successful applications in the early universe reconstruction problem [14,[28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Causality for Nonlocal Phenomena

Eckstein

Miller

2017

Ann. Henri Poincaré

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Abstract.Drawing from the theory of optimal transport we propose a rigorous notion of a causal relation for Borel probability measures on a given spacetime. To prepare the ground, we explore the borderland between Lorentzian geometry, topology and measure theory. We provide various characterisations of the proposed causal relation, which turn out to be equivalent if the underlying spacetime has a sufficiently robust causal structure. We also present the notion of the 'Lorentz-Wasserstein distance' and study its basic properties. Finally, we outline the possible applications of the developed formalism in both classical and quantum physics.

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The optimal mass transport problem for relativistic costs

Abstract: International audienc

Cited by 27 publications

References 29 publications

On the existence of dual solutions for Lorentzian cost functions

On the existence of dual solutions for Lorentzian cost functions

Kantorovich potentials and continuity of total cost for relativistic cost functions

Causality for Nonlocal Phenomena

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