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

点击翻译按钮获取中文摘要

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.