单生过程的研究进展  

Progress in single birth processes

作　　者：张余辉[1] Yuhui Zhang

机构地区：[1]北京师范大学数学科学学院,北京100875

出　　处：《中国科学：数学》2019年第3期621-642,共22页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11131003;11571043;11626245;11771047和11871008)资助项目

摘　　要：单生过程,作为最简单的过程—生灭过程的自然推广,本身具有可靠的研究背景.一方面,单生过程是对经典问题可预期有显式判别准则的最大类过程,从而使得单生过程成为研究无穷维反应扩散过程的基本工具;另一方面,单生过程通常是非对称的,因此被视为非对称过程的突出代表,而关于非对称过程的研究,至今所得结果并不完善,但对于单生过程,研究成果相对完整.本文将介绍单生过程经典问题的判别准则,包括唯一性、常返性、遍历性和强遍历性,还包括回返(灭绝)概率和平稳分布的表示,特别是近年来,我们得到了关于单生过程Poisson方程解的表示,以其为工具,举例说明如何统一处理上述问题以及回返时的多项式阶矩、指数阶矩和生命时的分布等.此外,本文还给出单生过程指数遍历的一个显式充分条件.最后回顾单生过程在粒子系统等模型研究中的一些具体应用.The single birth process,as a natural extension of birth and death process which is the simplest Q-process,has its own origins in research background.On one hand,the single birth process is nearly the largest class for which the explicit criteria on classical problems can be expected.Thus it becomes 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 are relatively 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 Poisson’s 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 ones of 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 some concrete applications of single birth processes in the study of particle systems and other models.

关 键 词：单生过程 生灭过程 遍历性 

分 类 号：O211.62[理学—概率论与数理统计]