Entropy, Pressure and Duality for Gibbs plans in Ergodic Transport

Abstract

Let $X$ be a finite set and $Ω=\{1,...,d\}^{\mathbb{N}}$ be the Bernoulli space. Denote by $σ$ the shift map acting on $Ω$. For a fixed probability $μ$ on $X$ with supp($μ$)$=X$, define $Π(μ,σ)$ as the set of all Borel probabilities $π\in P(X\times Ω)$ such that the $x$-marginal of $π$ is $μ$ and the $y$-marginal of $π$ is $σ$-invariant. We consider a fixed Lipschitz cost function $c: X \times Ω\to \mathbb{R}$ and an associated Ruelle operator. We introduce the concept of Gibbs plan, which is a probability on $X \times Ω$. Moreover, we define entropy, pressure and equilibrium plans. The study of equilibrium plans can be seen as a generalization of the optimal cost problem where the concept of entropy is introduced. We show that an equilibrium plan is a Gibbs plan. Our main result is a Kantorovich duality Theorem on this setting. The pressure plays an important role in the establishment of the notion of admissible pair. Finally, given a parameter $β$, which plays the role of the inverse of temperature, we consider equilibrium plans for $βc$ and its limit $π_\infty$, when $β\to \infty$, which is also known as ground state. We compare this with other previous results on Ergodic Transport in temperature zero.