Existence, duality, and cyclical monotonicity for weak transport costs
2019; Springer Science+Business Media; Volume: 58; Issue: 6 Linguagem: Inglês
10.1007/s00526-019-1624-y
ISSN1432-0835
AutoresJulio Backhoff‐Veraguas, Mathias Beiglböck, Gudmund Pammer,
Tópico(s)Point processes and geometric inequalities
ResumoAbstract The optimal weak transport problem has recently been introduced by Gozlan et al. (J Funct Anal 273(11):3327–3405, 2017). 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 and Juillet (On a mixture of brenier and strassen theorems. arXiv:1808.02681 , 2018) to general probability measures on $$\mathbb {R}^d$$ 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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