Entropy, Pressure and Duality for Gibbs plans in Ergodic Transport

Abstract

Let $X$ be a finite set and $\Omega=\{1,...,d\}^{\mathbb{N}}$ be the Bernoulli space. Denote by $\sigma$ the shift map acting on $\Omega$. For a fixed probability $\mu$ on $X$ with supp($\mu$)$=X$, define $\Pi(\mu,\sigma)$ as the set of all Borel probabilities $\pi \in P(X\times \Omega)$ such that the $x$-marginal of $\pi$ is $\mu $ and the $y$-marginal of $\pi$ is $\sigma$-invariant. We consider a fixed Lipschitz cost function $c: X \times \Omega \to \mathbb{R}$ and an associated Ruelle operator. We introduce the concept of Gibbs plan, which is a probability on $X \times \Omega$. 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 $\beta$, which plays the role of the inverse of temperature, we consider equilibrium plans for $\beta c$ and its limit $\pi_\infty$, when $\beta \to \infty$, which is also known as ground state. We compare this with other previous results on Ergodic Transport in temperature zero.