Duality Theorems in Ergodic Transport

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 $σ$ 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 $Π$ the set of all Borel probabilities $π$ on $X\times X$, such that, both its $x$ and $y$ marginal are $σ$-invariant probabilities. We are interested in the optimal plan $π$ which minimizes $\int c d π$ among the probabilities on $Π$. 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 $π$ by convex combinations of plans such that the support projects in periodic orbits.