Linear transfers as minimal costs of dilations of measures in balayage order

TL;DR

This paper characterizes linear transfers between probability measures as minimal costs of measure dilations, extending balayage theory and connecting it to Choquet capacities and Kantorovich duality.

Contribution

It introduces a novel characterization of linear transfers via balayage and extends balayage theory to include cost-optimizing measure dilations, linking to Kantorovich operators.

Findings

Linear transfers can be characterized by balayage with respect to cones of functions.

General linear transfers extend balayage by incorporating cost functionals.

Kantorovich operators are identified as duals and are related to capacities.

Abstract

Linear transfers between probability distributions were introduced in [5,6] in order to extend the theory of optimal mass transportation while preserving the important duality established by Kantorovich. It is shown here that $\{0, +\infty\}$-valued linear transfers can be characterized by balayage of measures with respect to suitable cones of functions \`a la Choquet, while general linear transfers extend balayage theory by requiring the "sweeping out" of measures to optimize certain cost functionals. We study the dual class of Kantorovich operators, which are natural and manageable extensions of Markov operators. It is also an important subclass of capacities, and could be called "convex functional Choquet capacities," since they play for non-linear maps the same role that convex envelopes do for arbitrary numerical functions. A forthcoming paper [7] will study their ergodic properties and their applications.