分支过程及相关模型的脊柱分解与应用  

Spine Decomposition and Applications of Branching Processes and Related Models

作　　者：刘荣丽[1] 任艳霞[2] 

机构地区：[1]南京大学数学系,南京江苏210093 [2]北京大学数学科学学院,北京100871

出　　处：《数学进展》2014年第2期183-205,共23页Advances in Mathematics（China）

基　　金：国家自然科学基金(No.11271030;No.11128101);江苏省自然科学基金(No.BK20130545)

摘　　要：分支过程、分支马氏过程和超过程在过去的几十年里越来越受关注,这是因为这类过程一方面与生物学有着紧密的联系,另一方面与数学的其他分支也有着紧密的联系.这类过程性质的研究最早主要是以分析的方法为主,近十几年来很多研究人员开始用概率的方法来研究这类过程的性质,其中一类重要的工具是脊柱方法.脊柱方法主要是依赖于测度的鞅变换和过程的脊柱分解,将上面三类过程中的随机多个轨道性质的研究转化成一个轨道性质的研究,从而大大简化了研究的内容,并且使证明变得非常直观.本文将介绍这种脊柱方法以及在三类过程中的一些应用.Due to their close connections to both biology and other branches of mathe- matics, branching processes, branching Markov processes and superprocesses have been receiving more and more attention. Early studies of these processes are mainly based on analytical meth- ods. In recent years, many people started to use probabilistic methods to study these processes. One of these probabilistic methods is the so-called spine method. The spine method mainly uses martingale transforms of measures and spine decomposition of the processes to reduce the study of a random number of sample paths of these processes to the study of one sample path. The spine method greatly simplifies the arguments, and makes the arguments more intuitive. In this paper, we give a survey of the spine method and its applications to the three types of processes above.

关 键 词：分支过程 鞅变换 分支马氏过程 超过程 行波解 

分 类 号：O211.65[理学—概率论与数理统计]