Decomposition of the Kantorovich problem and Wasserstein distances on simplexes

TL;DR

This paper shows that the Kantorovich metric on the set of T-invariant measures can be reconstructed from extreme points, and invariant optimal plans are mixtures of plans between these extreme points, with generalizations to constrained cases.

Contribution

It establishes a decomposition of the Kantorovich problem on invariant measure simplexes, linking the metric to extreme points and extending to constrained and ergodic decomposable simplexes.

Findings

Kantorovich metric can be reconstructed from extreme points.

Invariant optimal plans are mixtures of plans between extreme points.

Results extend to constrained Kantorovich problems and ergodic decomposable simplexes.

Abstract

Let $X$ be a Polish space, $\mathcal{P}(X)$ be the set of Borel probability measures on $X$, and $T\colon X\to X$ be a homeomorphism. We prove that for the simplex $\mathrm{Dom} \subseteq \mathcal{P}(X)$ of all $T$-invariant measures, the Kantorovich metric on $\mathrm{Dom}$ can be reconstructed from its values on the set of extreme points. This fact is closely related to the following result: the invariant optimal transportation plan is a mixture of invariant optimal transportation plans between extreme points of the simplex. The latter result can be generalized to the case of the Kantorovich problem with additional linear constraints and the class of ergodic decomposable simplexes.