Studies on recurrence and moments of transition probabilities of jump type Markov processes

本研究的主要成果如下：给出了无跳变Levy过程命中时间过程的Levy测度用局部时间表示。这个结果回答了一个问题：连续局部时间对不连续局部时间的自然推广是什么？虽然结果很简单，证明也很简短，但结果似乎是新的。此外，在转移概率为离散或绝对连续的情况下，我们分别用转移概率的概率函数或正则密度来表示命中时间过程的Levy测度。从存储过程的输入过程和释放速率的Levy测度出发，给出了存储过程重现性和暂态性的新的充分条件。应用这一结果，得到了当输入过程为稳定过程，释放速率为幂函数时，循环的充要条件。改进了已知的释放速率有界情况下的递归性和暂态性的充分条件。我们指出结果的一部分符合贝塞尔过程的递归-暂态条件。从输入过程的列维测度的广义矩的存在性（不存在性）出发，给出了存储过程广义矩存在（不存在）的条件。给出了存储过程转移概率的尾部行为与其输入过程利维测度的尾部行为之间的关系。为了得到这个结果，Chistyakov提出的次指数性概念和Rosinsky和Samorodnitky提出的次可加性概念是有用的。在2,3和4中常见的有趣现象是，当释放速率是幂函数时，Ornstein-UhLenbeck型过程起关键作用。

Main results of this research are the following :1.We gave representations of the Levy measureof the hitting time process of skip free Levy process in terms of its local time.This result is an answer to a question what is a natural generalization of the continuous local time to discontinuous one.While the result is simple and the proof is short, the result seems new.Moreover, we represented the Levy measure of the hitting time process by the probability function or the canonical density of the transition probability in case that the transition probability is discrete or absolutely continuous, respectively.2.We gave new sufficient conditions for recurrence and transience of storage process in terms of Levy measure of its input process and release rate.By applying this result we obtained the necessary and sufficient condition for recurrence in case thatthe input process is stable process and the release rate is a power function.We improved known sufficient conditions for recurrence and transience in case that the release rate is bounded.We pointed out that a part of the result corresponds to the recurrence-transience condition for Bessel processes.3.We gave conditions for the existence(nonexistence)of general ized moments of storage prooess in terms of the existence(nonexistence)of the generalized moments of the Levy measure of its input process.4.We gave relations between the tail behavior of the transition probability of storage process and the tail behavior of the Levy measure of its input process. In order to obtain this result, the concept subexponential ity, which was given by Chistyakov, and subadditivity, which is considered by Rosinsky and Samorodnitky, were useful.Interesting phenomenon common in 2, 3 and 4 is that Ornstein-UhLenbeck type process plays a critical role in case the release rate is a power function.