On the Starting Points of Incursions of Markov Processes
On the Starting Points of Incursions of Markov Processes
复制标题
马尔可夫过程入侵的起点
DOI：
10.1137/1113055
复制
发表时间：
1968
影响因子：
0.6
通讯作者：
A. Yushkevich
中科院分区：
文献类型：
--
作者：
E. Dynkin;A. Yushkevich
1.由Doob和Hunt提出的Martin边界理论的概率内容归结为一个事实，即马尔可夫过程的轨迹”开始”于入口边界点，”结束”于出口边界点。这一论断的第一部分比第二部分用复杂得多的术语明确地表述出来。这可以在几乎P链的Hunt理论（在随机时间生成，见[1]）的框架中完成，或者借助于随机时间变化（见[2]），或者最后，通过构造辅助时间非齐次马尔可夫过程（见[3]）。本文对马氏轨线的”起点”概念作了另一种解释，它适用于一般边界条件的研究.设xt是空间E中的马氏过程，Dc E. xt D分解为区间（或点）的连通分量的时间集。间隔不减少到点我们称之为入侵。本文证明了对于所有入侵的开始时刻，在入口边界拓扑中存在极限xr+ 0（对于终止于D的第一个出口的过程xt）。
1. The probabilistic content of the theory of Martin boundaries originatedby Doob and Hunt reduces to thefact that trajectories of a Markov process" start" at points of the entrance-boundary and" end" at exit-boundary points. The first part of thisassertion is explicitly formul-ated in much more complicated terms than the second. This can be done either in the framework of the Hunt theory of almost-P chains (generated at a random time, see [1]), or with the help of random time changes (see [2]), or, finally, by the construction of an auxiliary time inhomo-geneous Markov process (see [3]). In this article another interpretation is given for the idea of" starting points" of Markov trajectories which is suitable for the study of general boundary conditions.Let xt be a Markov process in the space E and let D c E. The set of times for which xt D decomposes into connected components which are intervals (or points). Theintervals not reducing to points we call incursions. In this note we prove that for the beginning times of all incursions there exist limits xr+ 0 in the entrance-boundary topology (for the processxt which terminates at the first exit from D).