Entropy, Pressure and Duality for Gibbs plans in Ergodic Transport

Let X be a finite set and Omega=1,...,d^mathbbN be the Bernoulli space. Denote by σ the shift map acting on Omega. For a fixed probability μ on X with supp(μ)=X, define Pi(μ,σ) as the set of all Borel probabilities π ∈ P(X× Omega) 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 × Omega → ℝ and an associated Ruelle operator. We introduce the concept of Gibbs plan, which is a probability on X × 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 β, which plays the role of the inverse of temperature, we consider equilibrium plans for β c and its limit π_∞, when β → ∞, which is also known as ground state. We compare this with other previous results on Ergodic Transport in temperature zero.