Duality Theorems in Ergodic Transport

TL;DR

This paper extends optimal transport theory to ergodic settings, establishing duality theorems, analyzing uniqueness, and exploring approximation methods for plans supported on periodic orbits.

Contribution

It introduces a duality framework for ergodic transport problems, generalizes classical results, and investigates the structure and approximation of optimal plans in ergodic contexts.

Findings

Established a Kantorovich duality theorem in ergodic transport.

Proved uniqueness of optimal plans under generic conditions.

Developed methods to approximate plans supported on periodic orbits.

Abstract

We analyze several problems of Optimal Transport Theory in the setting of Ergodic Theory. In a certain class of problems we consider questions in Ergodic Transport which are generalizations of the ones in Ergodic Optimization. Another class of problems is the following: suppose $\sigma$ is the shift acting on Bernoulli space $X=\{0,1\}^\mathbb{N}$, and, consider a fixed continuous cost function $c:X \times X\to \mathbb{R}$. Denote by $\Pi$ the set of all Borel probabilities $\pi$ on $X\times X$, such that, both its $x$ and $y$ marginal are $\sigma$-invariant probabilities. We are interested in the optimal plan $\pi$ which minimizes $\int c d \pi$ among the probabilities on $\Pi$. We show, among other things, the analogous Kantorovich Duality Theorem. We also analyze uniqueness of the optimal plan under generic assumptions on $c$. We investigate the existence of a dual pair of Lipschitz functions which realizes the present dual Kantorovich problem under the assumption that the cost is Lipschitz continuous. For continuous costs $c$ the corresponding results in the Classical Transport Theory and in Ergodic Transport Theory can be, eventually, different. We also consider the problem of approximating the optimal plan $\pi$ by convex combinations of plans such that the support projects in periodic orbits.