On A Mixture Of Brenier and Strassen Theorems

Gozlan, Nathael; Juillet, Nicolas

doi:10.48550/arxiv.1808.02681

Cited by 6 publications

(28 citation statements)

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“…Fundamental results in classical optimal transport theory include existence, duality, and cyclical monotonicity of optimizers. Through a series of contributions (see [34,2,32,6]) it has been understood that these results extend in full generality to the weak transport setup. We will recall these results in some detail in Section 2 below.…”

Section: Overviewmentioning

confidence: 82%

“…The optimal transport problem for weak costs was first introduced by Gozlan, Roberto, Samson and Tetali [34] and has immediately generated interest in several groups of researchers, see [45,46,29,32,1,2] among others. To present the basic problem we introduce some notation.…”

Section: Overviewmentioning

confidence: 99%

“…(Section 4.) • The Brenier-Strassen Theorem of Gozlan and Juillet [32] yields that 1-Lipschitz maps that are gradients to convex functions are the optimizers of transport problems with barycentric costs. This result plays a role in the probabilistic proof of the Caffarelli contraction theorem [28].…”

Section: Overviewmentioning

confidence: 99%

“…Indeed, particular costs of the form (OWT) were already considered by Marton [40,39] and Talagrand [48,49]. The theory for problem (OWT) has been further developed in [1,34,33,45,44,46,29,32,6,9,7]: Basic results of existence and duality are established in the articles [34,2,6]. The notion of C-monotonicity was developed in [4,32,6] as an analogue of classical c-cyclical montonicity in order to provide a characterization of optimizers to the weak transport problem.…”

Section: Literature Connected To the Weak Transport Problemmentioning

confidence: 99%

“…The theory for problem (OWT) has been further developed in [1,34,33,45,44,46,29,32,6,9,7]: Basic results of existence and duality are established in the articles [34,2,6]. The notion of C-monotonicity was developed in [4,32,6] as an analogue of classical c-cyclical montonicity in order to provide a characterization of optimizers to the weak transport problem. A weak transport analogue to the case of quadratic costs in classical optimal transport, is the case of barycentric costs.…”

Section: Literature Connected To the Weak Transport Problemmentioning

confidence: 99%

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Applications of weak transport theory

Backhoff-Veraguas¹,

Pammer²

2020

Preprint

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Motivated by applications to geometric inequalities, Gozlan, Roberto, Samson, and Tetali [34] introduced a transport problem for 'weak' cost functionals. Basic results of optimal transport theory can be extended to this setup in remarkable generality.In this article we collect several problems from different areas that can be recast in the framework of weak transport theory, namely: the Schrödinger problem, the Brenier-Strassen theorem, optimal mechanism design, linear transfers, semimartingale transport. Our viewpoint yields a unified approach and often allows to strengthen the original results.

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Section: Overviewmentioning

confidence: 82%

Section: Overviewmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

Section: Literature Connected To the Weak Transport Problemmentioning

confidence: 99%

Section: Literature Connected To the Weak Transport Problemmentioning

confidence: 99%

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Applications of weak transport theory

Backhoff-Veraguas¹,

Pammer²

2020

Preprint

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Existence, duality, and cyclical monotonicity for weak transport costs

2019

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The optimal weak transport problem has recently been introduced by Gozlan et. al. [25]. We provide general existence and duality results for these problems on arbitrary Polish spaces, as well as a necessary and sufficient optimality criterion in the spirit of cyclical monotonicity. As an application we extend the Brenier-Strassen Theorem of Gozlan-Juillet [23] to general probability measures on R d under minimal assumptions.A driving idea behind our proofs is to consider the set of transport plans with a new ('adapted') topology which seems better suited for the weak transport problem and allows to carry out arguments which are close to the proofs in the classical setup.

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Weak monotone rearrangement on the line

Backhoff-Veraguas¹,

Beiglböck²,

Pammer³

2020

Electron. Commun. Probab.

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Weak optimal transport has been recently introduced by Gozlan et al. The original motivation stems from the theory of geometric inequalities; further applications concern numerics of martingale optimal transport and stability in mathematical finance.In this note we provide a complete geometric characterization of the weak version of the classical monotone rearrangement between measures on the real line, complementing earlier results of Alfonsi, Corbetta, and Jourdain. arXiv:1902.05763v1 [math.PR]

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On A Mixture Of Brenier and Strassen Theorems

Cited by 6 publications

References 0 publications

Applications of weak transport theory

Applications of weak transport theory

Existence, duality, and cyclical monotonicity for weak transport costs

Weak monotone rearrangement on the line

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