Existence, duality, and cyclical monotonicity for weak transport costs
Existence, duality, and cyclical monotonicity for weak transport costs
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DOI：
10.1007/s00526-019-1624-y
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发表时间：
2019-12-01
影响因子：
2.1
通讯作者：
Pammer, G.
中科院分区：
文献类型：
--
作者：
Backhoff-Veraguas, J.;Beiglboeck, M.;Pammer, G.
Gozlan 等人最近提出了最优弱传输问题。 （功能分析杂志 273(11):3327-3405, 2017）。我们提供了这些问题在任意波兰空间上的一般存在性和对偶性结果，以及本着循环单调性的精神的充分必要的最优性准则。作为一种应用，我们将 Gozlan 和 Juillet 的 Brenier-Strassen 定理（On a mix of brenier and strassen theorems., 2018）扩展到最小假设下 Rd\ 上的一般概率度量。我们证明背后的一个驱动思想是考虑具有新的（“适应的”）拓扑的传输计划集，该拓扑似乎更适合弱传输问题，并且允许进行与经典设置中的证明接近的论证。
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. , 2018) to general probability measures on Rd\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.