R-positive Recurrence And Quasi-stationary Distributions For One-dimensional Diffusions

Today quasi-stationary distributions (in short, QSDs) of Markov processes havebeen one of the most active research area. A one-dimensional difusions will nowbe taken to mean a continuous strong Markov process with values in (0,∞), whoseQSDs and related issues hadn’t been researched with system and deep enough.In this paper, we mainly study R-positive recurrence, QSDs and their domain ofattraction problem for one-dimensional difusions.In the first chapter, the research background, the development course, the re-search status of R-positive recurrence and QSDs will be introduced. Also, the mainresearch results of this paper will be given.In the second chapter, we will introduce the basic theory, including one dimen-sional difusions and QSDs.In the third chapter, we study R-positive recurrence for one-dimensional difu-sions. We consider the one-dimensional difusion X killed at0, which the infinityis either a natural boundary or an entrance boundary in the sense of Feller. Wegive some sufcient conditions for R-positivity of the process killed at0. Theseconditions are only based on the drift, which are easy to check.In the fourth chapter, we study QSDs for one-dimensional difusions under thecondition that0is a regular boundary and∞is a natural boundary. when0isa regular boundary and∞a natural boundary, we not only give a necessary andsufcient condition for the existence of a QSD, but also we construct all QSDs forone-dimensional difusions.In the fifth chapter, we study QSDs for one-dimensional difusions under thecondition that0is a regular boundary and∞is an entrance boundary. when0isa regular boundary and∞is an entrance boundary, we will show that a necessaryand sufcient condition for the uniqueness of QSD, and that this unique QSD ν1attracts all initial distributions ν supported on (0,∞), as well as existence of theQ-process.