Generalized Orlicz spaces and Wasserstein distances for convex-concave scale functions

TL;DR

This paper introduces a generalized framework for Wasserstein distances and Orlicz spaces using convex-concave scale functions, expanding the mathematical tools for probability measures on metric spaces.

Contribution

It extends the concept of Wasserstein distances and Orlicz spaces to functions composed of convex and concave parts, including all twice differentiable functions.

Findings

Defines a new class of Wasserstein distances based on convex-concave functions.

Extends Orlicz space theory to include these generalized functions.

Provides a mathematical foundation for future applications in probability and analysis.

Abstract

Given a strictly increasing, continuous function $\vartheta:\R_+\to\R_+$, based on the cost functional $\int_{X\times X}\vartheta(d(x,y))\,d q(x,y)$, we define the $L^\vartheta$-Wasserstein distance $W_\vartheta(\mu,\nu)$ between probability measures $\mu,\nu$ on some metric space $(X,d)$. The function $\vartheta$ will be assumed to admit a representation $\vartheta=\phi\circ\psi$ as a composition of a convex and a concave function $\phi$ and $\psi$, resp. Besides convex functions and concave functions this includes all $\mathcal C^2$ functions. For such functions $\vartheta$ we extend the concept of Orlicz spaces, defining the metric space $L^\vartheta(X,m)$ of measurable functions $f: X\to\R$ such that, for instance, $$d_\vartheta(f,g)\le1\quad\Longleftrightarrow\quad \int_X\vartheta(|f(x)-g(x)|)\,d\mu(x)\le1.$$