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Duality Theorems in Ergodic Transport

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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={1,2,…,d}ℕ, and, consider a fixed continuous cost function c:X×X→ℝ. Denote by Π the set of all Borel probabilities π on X×X, such that, both its x and y marginals are σ-invariant probabilities. We are interested in the optimal plan π which minimizes ∫c  dπ among the probabilities in 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.

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Notes

  1. ν∈P(Γ) is invariant when ∫fdν=∫(f∘σ)dν for any f∈C(Γ).

  2. Our purpose here is to describe the Ergodic Transport Problem in a broad sense. In this way it is natural to consider invariance not only for σ, but, also for \(\hat{\sigma }\).

  3. We point out that g is a continuous linear functional over M(Y) and not necessarily a function from Y to ℝ.

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Authors and Affiliations

  1. Instituto de Matemática, UFRGS, Porto Alegre, Brazil

    Artur O. Lopes & Jairo K. Mengue

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  1. Artur O. Lopes
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  2. Jairo K. Mengue
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Correspondence to Artur O. Lopes.

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The first author partially supported by DynEurBraz, CNPq, PRONEX – Sistemas Dinamicos, INCT, Convenio Brasil-Franca.

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Lopes, A.O., Mengue, J.K. Duality Theorems in Ergodic Transport. J Stat Phys 149, 921–942 (2012). https://doi.org/10.1007/s10955-012-0626-3

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  • DOI: https://doi.org/10.1007/s10955-012-0626-3

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