Weak optimal transport with unnormalized kernels

TL;DR

This paper introduces a novel variant of weak optimal transport using unnormalized kernels, providing dual formulas, attainment conditions, and a new characterization of stochastic order related to convex functions.

Contribution

It develops a new weak optimal transport framework with unnormalized kernels, establishing duality results and stochastic order characterizations.

Findings

Dual formula for the new transport problem

Conditions for primal and dual attainment

Transport characterization of convex stochastic order

Abstract

We introduce a new variant of the weak optimal transport problem where mass is distributed from one space to the other through unnormalized kernels. We give sufficient conditions for primal attainment and prove a dual formula for this transport problem. We also obtain dual attainment conditions for some specific cost functions. As a byproduct we obtain a transport characterization of the stochastic order defined by convex positively 1-homogenous functions, in the spirit of Strassen theorem for convex domination.