单生过程的研究进展

Abstract

The single birth process, as a natural extension of birth and death processwhich is the simplest $Q$-process, has its own origins in research background. On one hand, the single birth processis nearly the largest class for which the explicit criteria on classical problems can be expected. Thus itbecomes a fundamental comparison tool in studying complicated processes, such as infinite-dimensional reaction-diffusion processes.On the other hand, the single birth process is usually non-symmetric and hence is regarded as a representative of the non-symmetric processes.For non-symmetric processes, in contrast to symmetric ones, our knowledge is very limited, except for single birth processes to which many results arerelatively completed. In this survey paper, we present some criteria on classical problems of single birth processes, including uniqueness, recurrence, ergodicity and strong ergodicity, as well as representation of return(or extinction) probability and stationary distribution. Recently explicit representation of solution of Poissons equation has been obtained. Based on this tool,a unified treatment of various problems for general single birth processes is presented. To illustrate this approach we deal with uniqueness, recurrence and return probability as examples. Polynomial moments and exponential onesof return time as well as the distribution of explosion time are obtained similarly. Moreover, we present an explicit and sufficient condition for exponential ergodicity of single birth processes. At last, we review someconcrete applications of single birth processes in the study of particle systems and other models.