The Last Exit Time Of Symmtrical Markov Processes And Its Related Problems

The first hitting time and the last exit time play very basic and important roles in the research of Markov processes. The last exit time is a useful tool to study the behavior of the processes. We can visually define the recurrence and transience for sets and processes by the last exit time. There are many elementary applications of the last exit time in " probabilitistic potential thoery ", especially for the studying of equilibrium potential. The investigation of last exit times of Markov processes has a long history. Many scholars have done a large amount of work on the last exit for some special class of processes, for example, Brownian motion, symmetric stable process, Levy process and so on. And many papers(even some recent developments), which study the path properties of processes such as asymptotic behavior and escape rates, depend upon using the last exit time as an important tool. Now, the researches for Brownian motion are relatively abundant and perfect. However, these results are almost for Levy procesess, which have independent increments. It is difficult to obtain the similar results for the general processes without the property of independ increments. In this paper, we want to extend the work of Brownian motion to some general processes, which have continuous sample paths but not independent increments. We begin our work with elliptic diffusion process that is a Hunt process in the sense of Blumenthal and Getoor [3] and has continuous sample paths. The elliptic diffusion is an important and special class of Markov processes. We can find many research results about the elliptic diffusion, which can show that the elliptic diffusion has many similar properties with Brownian motion. Notice that the elliptic diffusion has not independent increments and can be bounded by two (non-standard) Brownian motions, we can say that the elliptic diffusion is a bridge from Brownian motions to general diffusion processes.Our paper also contains a party of probabilistic potential thoery. The potential thoery has a long history. The classical potential thoery came into being in the middle period of 19th century. It has a strong physical background and is the most fundamental mathematical tool of electromagnetics. In the 20th century, the potential theory developed rapidly, especially after people found the connection between Brownian motion and classical potential theory. The important and beautiful connection was first investigated by Kakutani,Kac and Doob. Hunt then showed that the potential theory could be developed by probabilistic methods for a large class of transient Markov processes. On the other hand, the potential tools(analytic tools) also accelerated the development of the probability theory by a long way. Then probabilistic potential thoery, a lifeful and enrichment field of mathematics, cameinto being. It provided a weath of further information on both the analytic and prob-abilitic aspects. In our paper the emphasis is on equilibrium potential. Equilibrium potential is one of the most famous three problems in classical potential theory (the other two are Dirichlet problem and Balayage problem). The equilibrium problem is related to the last exit time closely. Chung[9] (1973) established a relationship between last exit distribution and equilibrium measure for Markov processes. In spite of the large amount of written material on probabilistic potential theory and on the connection between Brownian motion and classical potential, the fine results about general processes cannot be found easily. In our paper, we consider two fundamental conceptions of probabilistic potential theory: regularity and transience. We extend some fine results from Brownian motion to the elliptic diffusion process.Our work is composed of four parts.Finally, we obtain the estimation of the distribution of the maximum excursion for elliptic diffusion process, then the sufficient and necessary condition for the existence of the k-moments of the maximum excursion is k + 2 < d. All these facts show that elliptic diffusion h...