Existence, Duality, and Cyclical monotonicity for weak transport costs
Abstract
The optimal weak transport problem has recently been introduced by Gozlan et.\ al. 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 to general probability measures on 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.
Keywords
Cite
@article{arxiv.1809.05893,
title = {Existence, Duality, and Cyclical monotonicity for weak transport costs},
author = {Julio Backhoff Veraguas and Mathias Beiglboeck and Gudmund Pammer},
journal= {arXiv preprint arXiv:1809.05893},
year = {2019}
}
Comments
We credit a new version of the work by Gozlan-Juillet where they extend their Brenier-Strassen Theorem to the non-compact case. We improved our proof of this same result to make a new self-contained presentation. A number of typos and clarifications were also amended