论文标题
关于随机微分方程的本地混合技术的注意
Note on local mixing techniques for stochastic differential equations
论文作者
论文摘要
本文讨论了几种技术，这些技术可用于将耦合方法应用于随机微分方程（SDE）的解决方案。它们都在尺寸$ d \ ge 1 $中工作，尽管在$ d = 1 $中，最自然的方法是使用轨迹的交叉点，这仅需要强大的马尔可夫属性和扩散系数的非分类。在尺寸$ d> 1 $中，可以通过考虑离散时间$ n = 0,1，\ ldots $或安排特殊停止时间序列并使用本地马可福音-Dobrushin's（MD）条件来使用嵌入式Markov链。进一步的应用程序可能基于一个或另一个版本的MD条件。对于收敛和混合速率的研究（马尔可夫）过程必须是强大的马尔可夫和经常性。但是，复发是一个单独的问题，本文未讨论。
This paper discusses several techniques which may be used for applying the coupling method to solutions of stochastic differential equations (SDEs). They all work in dimension $d\ge 1$, although, in $d=1$ the most natural way is to use intersections of trajectories, which requires nothing but strong Markov property and non-degeneracy of the diffusion coefficient. In dimensions $d>1$ it is possible to use embedded Markov chains either by considering discrete times $n=0,1,\ldots$, or by arranging special stopping time sequences and to use local Markov -- Dobrushin's (MD) condition. Further applications may be based on one or another version of the MD condition. For studies of convergence and mixing rates the (Markov) process must be strong Markov and recurrent; however, recurrence is a separate issue which is not discussed in this paper.
