遍历输运中的对偶定理
Duality Theorems in Ergodic Transport
摘要：
本文在遍历理论的背景下分析了最优运输理论的几个问题。在某一类问题中，我们考虑遍历运输中的问题，这些问题是遍历优化中的问题的推广。另一类问题如下：假设$σ$是作用在伯努利空间$X=\{0,1\}^\mathbb{N}$上的移位，并且，考虑固定的连续成本函数$c:X \times X\to \mathbb{R}$。用$Π$表示$X\times X$上所有Borel概率$π$的集合，使得其$x$和$y$边际都是$σ$不变概率。我们感兴趣的最佳计划$π$，最小化$\int c d π$之间的概率$Π$。我们显示，除其他事项外，类似的康托洛维奇对偶定理。我们还分析了在一般假设下的$c$的最优计划的唯一性。我们研究的存在性的对偶Lipschitz函数实现目前的对偶Kantorovich问题的假设下，成本是Lipschitz连续的。对于连续成本$c$，经典输运理论和遍历输运理论的相应结果最终可能不同。我们还考虑了用凸规划组合逼近最优规划$π$的问题，使得支撑投影在周期轨道上。
英文摘要：
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 $X=\{0,1\}^\mathbb{N}$, and, consider a fixed continuous cost function $c:X \times X\to \mathbb{R}$. Denote by $Π$ the set of all Borel probabilities $π$ on $X\times X$, such that, both its $x$ and $y$ marginal are $σ$-invariant probabilities. We are interested in the optimal plan $π$ which minimizes $\int c d π$ among the probabilities on $Π$. 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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