椭圆扩散马氏最大耦合的刚性
Rigidity for Markovian Maximal Couplings of Elliptic Diffusions
摘要：
最大耦合是马氏过程的概率耦合，使得耦合时刻的尾概率一致地达到全变差下界（Aldous界）.马尔可夫（或浸入）耦合是由策略定义的耦合，其中任何一个过程都不允许在进行下一次转换之前查看另一个过程的未来。马尔可夫耦合通常比一般耦合更容易构造和分析，并且在概率和分析的许多分支中发挥重要作用。Hsu和Sturm（2013）证明了布朗运动的反射耦合是从两个不同点出发的布朗运动的唯一马尔可夫最大耦合（MMC）。后来，Kuwada（2009）证明了黎曼流形上布朗运动的MMC的存在性强制了流形上反射结构的存在性。在这项工作中，我们调查适当的经常椭圆形扩散流形上，并显示如何考虑的扩散几何（包括尺寸的等距群和流量的等距）是根本的分类的空间和发电机的扩散MMC存在，特别是当MMC也持有下局部扰动的起点耦合扩散。我们还描述了这样的扩散的Killing向量场（等距群的生成器）和膨胀向量场（缩放对称群的生成器）。
英文摘要：
Maximal couplings are (probabilistic) couplings of Markov processes such that the tail probabilities of the coupling time attain the total variation lower bound (Aldous bound) uniformly for all time. Markovian (or immersion) couplings are couplings defined by strategies where neither process is allowed to look into the future of the other before making the next transition. Markovian couplings are typically easier to construct and analyze than general couplings, and play an important role in many branches of probability and analysis. Hsu and Sturm (2013) proved that the reflection-coupling of Brownian motion is the unique Markovian maximal coupling (MMC) of Brownian motions starting from two different points. Later, Kuwada (2009) proved that the existence of a MMC for Brownian motions on a Riemannian manifold enforces existence of a reflection structure on the manifold. In this work, we investigate suitably regular elliptic diffusions on manifolds, and show how consideration of the diffusion geometry (including dimension of the isometry group and flows of isometries) is fundamental in classification of the space and the generator of the diffusion for which an MMC exists, especially when the MMC also holds under local perturbations of the starting points for the coupled diffusions. We also describe such diffusions in terms of Killing vectorfields (generators of isometry groups) and dilation vectorfields (generators of scaling symmetry groups).
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