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 $\int c d \pi$ 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

On the Monge–Kantorovich problem with additional linear constraints

“…Another example of interesting linear restrictions is the invariance with respect to a continuous action of some group. Such problems can naturally appear in the ergodic theory (see, for example, [11]) or geometry ( [12]). If the cost function is invariant, it is known that solutions of the classical Monge-Kantorovich plans are also invariant [12].…”

Ergodic and thermodynamic games

“…Here, we will introduce a cooperative game. We will use tools from ergodic transport (see [5]). As we said in the introduction, we will not elaborate on this example.…”

“…This means that π has to have x-projection given by µ. In other words, π needs to belong to ∈ M µ,S (X × Y ), which is the set of plans whose x-projection is µ and y-projection is invariant for S (see [5]).…”

“…Ergodic transport was introduced recently, see for example [5,6] and among other applications, we can find the invariant measure that minimizes the Wasserstein-2 distance (see [13]) from a given measure which is not invariant: we just have to use as A 2 the cost function given by the square of the distance, and search for the measure that minimizes (instead of maximize) the expression above. In this case, µ will play the role of any given measure, not necessarily invariant.…”

“…We present some examples in the text and also discuss why uniqueness of solutions is not a trivial question. We finish with a different model (the hierarchical case), that is a special case without entropy, that allow us to use some concepts of cooperative games leading to the use of ergodic transport techniques (see [5,6]). We stress, however, the fact that this example is unique in the text: all other situations considered here, including the definition of Nash equilibrium and proofs of existence, are in the domain of non-cooperative ergodic game theory, where players are supposed not to communicate or collaborate with each other.…”