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

本文的主要研究结果如下：1.给出了无跳过Levy过程的打中时间过程的Levy测度的局部时间表示。这一结果回答了连续局部时间到间断局部时间的自然推广是什么的问题。虽然结果简单且证明简短，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，我们用转移概率的概率函数或转移概率的典范密度来表示击中时间过程的Levy测度。2.利用输入过程和释放率的Levy测度，给出了存储过程的常返性和暂时性的新的充分条件。应用这一结果，得到了输入过程为稳定过程，释放率为幂函数时常返性的充要条件。在释放率有界的情况下，改进了已知的常返性和暂时性的充分条件。指出了部分结果对应于Bessel过程的常返性-瞬时性条件。3.给出了一般存储过程矩存在(不存在)的条件(不存在)其输入过程的Levy测度的广义矩。4.给出了存储过程转移概率的尾态与其输入过程的Levy测度的尾态之间的关系。为了得到这一结果，Chistyakov给出的次指数性的概念和Rosinsky和Samorodnitky考虑的次可加性的概念是有用的。有趣的是，在释放率为幂函数的情况下，Ornstein-Uhlenbeck型过程在释放率为幂函数的情况下起着关键作用。

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.