Guasi-ergodicity Of Absorbing Markov Processes And Related Topics

In real life, people are often faced with stochastic systems with certain loss of probability but a steady trend, however, they often do not have usual ergod-icity. People are concerned mainly with the asymptotic behavior of such systems conditioned on long-term survival. The concept of quasi-stationarity and quasi-ergodicity can capture the long time behavior of the system conditioned on non-extinction. The purpose of this thesis is to study guasi-ergodicity of absorbing Markov processes and related topics. We study the existence, uniqueness and domain of attraction of the quasi-ergodic distribution and the quasi-stationary distribution, and we also give a comparison between the quasi-ergodic distribu-tion and the quasi-stationary distribution. The thesis is divided into five chapters as follows.In Chapter 1,we introduce the research background where the problems are produced, and the research progress of quasi-stationary distribution and quasi-ergodic distribution of absorbing Markov processes. Also, we introduce the spe,cific model of this thesis , review the basic definitions in this thesis and list the organization of the paper and the research train of thought.In Chapter 2, we study the quasi-ergodic distribution of absorbing Markov processes. We give a sufficient condition for the existence and uniqueness of a quasi-ergodic distribution for absorbing Markov processes. We also show that the unique quasi-ergodic distribution is stochastically larger than the unique quasi-stationary distribution in the sense of monotone likelihood-ratio ordering for the process. Using an orthogonal-polynomial approach, we prove that these results are valid for the birth-death process on the nonnegative integers with 0 an absorbing boundary and +? an entrance boundary. Moreover , we give an alternative proof of eigentime identity for the birth-death process with exit boundary. A note on the quasi-ergodic distribution of Markov processes with fast return from infinity is also given. Under suitable assumptions, we prove that there exists a unique quasi-ergodic distribution for Markov processes with fast return from infinity.In Chapter 3, we study the quasi-ergodic distribution for one-dimensional dif-fusions killed at 0, when 0 is an exit boundary and +? is an entrance boundary.Using the spectral theory tool, we show that if the killed semigroup is intrinsically ultracontractive, then there exists a unique quasi-ergodic distribution for the pro-cess. An example is given to illustrate the result. Moreover, the ultracontractivity of the killed semigroup is also studied. We give a necessary and sufficient criterion for the killed semigroup is ultracontractive.In Chapter 4,we study quasi-stationary distributions for one-dimensional diffusions killed at 0, when 0 is a regular boundary and +? is an inaccessible boundary. We give a necessary and sufficient condition for the existence of a quasi-stationary distribution for the one-dimensional diffusions. When +? is a natural boundary, we prove that there is an infinite continuum of quasi-stationary distributions for the process,and we also construct all quasi-stationary distribu-tions. Moreover, we give a sufficient condition for R-positivity of the process killed at the origin. This condition is only based on the drift, which is easy to check.When +? is an entrance boundary , We give a necessary and sufficient condition for the existence of exactly one quasi-stationary distribution, and we also showthat this distribution attracts all initial distributions.In Chapter 5,we summarize the results and innovative points of this the-sis , and also present some interesting questions for further research.