Extending Markov Processes in Weak Duality by Poisson Point Processes of Excursions
Extending Markov Processes in Weak Duality by Poisson Point Processes of Excursions
复制标题
DOI：
10.1007/978-3-540-70847-6_7
复制
发表时间：
2007
期刊：
影响因子：
--
通讯作者：
Zhen-Qing Chen;M. Fukushima;J. Ying
中科院分区：
文献类型：
--
作者：
Zhen-Qing Chen;M. Fukushima;J. Ying
设 a 为拓扑空间 E 的非孤立点。假设我们在 E0= E\{a} 上得到标准过程 X 0 和 ̂X 0 ，其弱对偶性相对于 E0 上的 σ 有限测度 m，它们在 E0 内没有杀戮，但可接近 a。我们首先表明，它们的扩展 X 和 ̂X 到 E 不允许在 a 处停留并保持弱对偶性，这是由 X 0 、 ̂X 0 和 m 接近代表 X 在 a 处的杀灭率的非负常数 δ0 的概率唯一确定的。然后，我们从 X 0 开始，通过将围绕 a 的返回偏移和包括即时杀戮在内的可能的非返回偏移拼凑在一起来构造这样的 X。这扩展了 M. Fukushima 和 H. Tanaka [16] 最近的结果，该结果处理 X 0, X 是 m 对称扩散且 X 不允许在 a 处逗留或杀戮的情况。论文最后给出了欧氏域上跳跃型对称马尔可夫过程和非对称扩散的典型例子。
Let a be a non-isolated point of a topological space E. Suppose we are given standard processes X 0 and ̂X 0 on E0= E\{a} in weak duality with respect to a σ-finite measure m on E0 which are of no killings inside E0 but approachable to a. We first show that their extensions X and ̂X to E admitting no sojourn at a and keeping the weak duality are uniquely determined by the approaching probabilities of X 0, ̂X 0 and m up to a non-negative constant δ0 representing the killing rate of X at a. We then construct, starting from X 0, such X by piecing together returning excursions around a and a possible non-returning excursion including the instant killing. This extends a recent result by M. Fukushima and H. Tanaka [16] which treats the case where X 0, X are m-symmetric diffusions and X admits no sojourn nor killing at a. Typical examples of jump type symmetric Markov processes and non-symmetric diffusions on Euclidean domains are given at the end of the paper.