迭代函数系统的最优迁移与平稳测度
Optimal transportation and stationary measures for Iterated Function Systems
摘要：
在这篇文章中，我们展示了如何从最优运输的思想，方法和结果可以用来研究配备了概率分布的迭代函数系统的平稳measures的各个方面。在平均收缩假设下，我们恢复了一个经典的存在唯一性结果，证明了可以导出尾估计的广义矩界，考虑了相关马尔可夫链的经验测度的收敛性，并证明了在许多情况下，当系统扰动时，平稳测度的Lipschitz连续性，其结果是几乎在扰动的每个参数处都具有“线性响应公式”。
英文摘要：
In this article we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measuresof Iterated Function Systems equipped with a probability distribution. We recover a classical existence and uniqueness result under a contraction-on-average assumption, prove generalized moment bounds from which tail estimates can be deduced, consider the convergence of the empirical measure of an associated Markov chain, and prove in many cases the Lipschitz continuity of the stationary measure when the system is perturbed, with as a consequence a "linear response formula" at almost every parameter of the perturbation.
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