Optimal transportation of processes with infinite Kantorovich distance. Independence and symmetry

TL;DR

This paper investigates optimal transportation mappings for probability measures on infinite-dimensional spaces under the Kantorovich distance, focusing on symmetric, exchangeable, and stationary measures, and links existence to ergodic properties.

Contribution

It establishes existence results for symmetric optimal transportation in infinite dimensions, especially for stationary Gibbs measures, and explores the role of ergodicity.

Findings

Existence of symmetric optimal transportation for certain stationary Gibbs measures.

Connection between transportation existence and ergodicity of the target measure.

Analysis of measures with symmetric properties like exchangeability and stationarity.

Abstract

We consider probability measures on $\mathbb{R}^{\infty}$ and study optimal transportation mappings for the case of infinite Kantorovich distance. Our examples include 1) quasi-product measures, 2) measures with certain symmetric properties, in particular, exchangeable and stationary measures. We show in the latter case that existence problem for optimal transportation is closely related to ergodicity of the target measure. In particular, we prove existence of the symmetric optimal transportation for a certain class of stationary Gibbs measures.