Existence, Duality, and Cyclical monotonicity for weak transport costs

TL;DR

This paper establishes foundational existence, duality, and optimality criteria for weak transport costs on Polish spaces, extending classical results like Brenier-Strassen to broader settings.

Contribution

It introduces a new topology for transport plans and extends key theorems to general probability measures under minimal assumptions.

Findings

Proved existence and duality for weak transport problems on Polish spaces.

Derived a necessary and sufficient optimality criterion based on cyclical monotonicity.

Extended the Brenier-Strassen Theorem to general probability measures on R^d.

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 $R^d$ 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.