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 $\sigma$ 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 $\int cd\pi$ 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.

References in corpus (4)

• On the zero-temperature limit of Gibbs states
• Entropy and Variational Principle for one-dimensional Lattice Systems with a general a-priori probability: positive and zero temperature
• Ergodic Transport Theory, periodic maximizing probabilities and the twist condition
• The effective potential and transshipment in thermodynamic formalism at temperature zero

Cited by in corpus (8)

• Entropy and Variational Principle for one-dimensional Lattice Systems with a general a-priori probability: positive and zero temperature
• Ergodic optimization, zero temperature limits and the max-plus algebra
• Optimal Transportation Theory with Repulsive Costs
• Optimal transportation of processes with infinite Kantorovich distance. Independence and symmetry
• On the Monge-Kantorovich problem with additional linear constraints
• A Strong Duality Principle for Equivalence Couplings and Total Variation
• Entropy, Pressure and Duality for Gibbs plans in Ergodic Transport
• An Ergodic Theorem on Ergodic Transport