耦合Kolmogorov扩散：极大性和效率考虑
Coupling the Kolmogorov Diffusion: maximality and efficiency considerations
摘要：
这是一个案例研究概率耦合的幂零扩散发生率。我们专注于最简单的情况下Kolmogorov扩散（布朗运动连同其时间积分，或者，稍微更一般地说，连同有限数量的迭代时间积分）。在这种情况下，不可能有马尔可夫最大耦合。实际上，马尔可夫耦合甚至不能是有效的（扩展Burdzy和Kendall的术语，Efficient Markovian coupling：examples and counterexamples; Annals of Applied Probability，2000）。最后，至少在单时间积分的经典情况下，不可能选择一个马尔可夫耦合，这是最佳的意义上，同时最小化失败的概率耦合的时间t为所有积极的t。为了补偿所有这些负面的结果，我们表现出一个简单有效的非马尔可夫耦合。
英文摘要：
This is a case study concerning the rate at which probabilistic coupling occurs for nilpotent diffusions. We focus on the simplest case of Kolmogorov diffusion (Brownian motion together with its time integral, or, slightly more generally, together with a finite number of iterated time integrals). In this case there can be no Markovian maximal coupling. Indeed, Markovian couplings cannot even be efficient (extending the terminology of Burdzy and Kendall, Efficient Markovian couplings: examples and counterexamples; Annals of Applied Probability, 2000). Finally, at least in the classical case of a single time integral, it is not possible to choose a Markovian coupling that is optimal in the sense of simultaneously minimizing the probability of failing to couple by time t for all positive t. In recompense for all these negative results, we exhibit a simple efficient non-Markovian coupling.
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