Characterization of a class of weak transport-entropy inequalities on the line

TL;DR

This paper investigates a specific weak transport cost related to convex order on the real line, providing conditions for weak transport-entropy inequalities and a form of the convex Poincaré inequality in one dimension.

Contribution

It characterizes a class of weak transport-entropy inequalities on the real line and identifies a coupling that optimizes the weak transport cost independently of the cost function.

Findings

Weak transport cost is achieved by a cost-independent coupling.

Provides necessary and sufficient conditions for weak transport-entropy inequalities in dimension one.

Derives a weak transport-entropy form of the convex Poincaré inequality.

Abstract

We study an optimal weak transport cost related to the notion of convex order between probability measures. On the real line, we show that this weak transport cost is reached for a coupling that does not depend on the underlying cost function. As an application, we give a necessary and sufficient condition for weak transport-entropy inequalities in dimension one. In particular, we obtain a weak transport-entropy form of the convex Poincar{\'e} inequality in dimension one.