The Genealogy of Extremal Particles of Branching Brownian Motion
The Genealogy of Extremal Particles of Branching Brownian Motion
复制标题
分支布朗运动极值粒子谱系
DOI：
10.1142/9789811206092_0004
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
K. Saha
中科院分区：
文献类型：
--
作者：
S. Kliem;K. Saha
分支布朗运动（BBM）被用作种群模型，其中粒子在真实的线上的位置表示他或她的适应度，该适应度根据由于突变而引起的布朗运动而演化。在这种情况下，粒子的谱系树是重要的理解随着时间的推移人口的演变。本文的主要目标是给一个概述和动机的一些结果的谱系的极值粒子BBM。在本文的第一部分，我们讨论的结果L。- P. Arguin，A. Bovier和N.奇石来自[Genealogy of Extremal Particles of BBM，2011]和[Poissonian Statistics in the Extremal Process of BBM，2012]。他们的文章给出了BBM极限极值过程分布的部分答案。这两篇文章着重介绍了如何利用遗传信息成功地获得关于极限粒子图象的相当广泛的信息。Berestycki和J. Schweinsberg来自[吸收BBM的谱系，2013]。具有吸收的BBM与具有选择的种群模型有关。这篇文章表明，家谱的幸存粒子（具有选择性的优势），这是“本地”的极值粒子，然后由Bolthausen-Sznitman聚结过程。这一结果与没有选择的标准种群模型形成鲜明对比。
Branching Brownian motion (BBM) is used as a population model where the location of a particle on the real line represents his or her fitness which evolves according to a Brownian motion due to mutations. In this context, the genealogical tree of the particles is important to understand the evolution of the population over time.The main goal of this article is to give an overview and motivations for some of the results on the genealogy of extremal particles of BBM.In the first part of this article, we discuss the results of L.-P. Arguin, A. Bovier and N. Kistler from [Genealogy of Extremal Particles of BBM, 2011] and [Poissonian Statistics in the Extremal Process of BBM, 2012]. Their articles give a partial answer to what the distribution of the limiting extremal process of BBM looks like. These two articles highlight, how genetical information can successfully be used to obtain rather extensive information for the limiting particle picture.In the second part, we discuss the results of J. Berestycki, N. Berestycki and J. Schweinsberg from [The Genealogy of BBM with Absorption, 2013]. BBM with absorption is related to a population model with selection. This article shows that the genealogy of the surviving particles (with a selective advantage), which are ‘local’ extremal particles, is then governed by a Bolthausen-Sznitman coalescent process. This result is in sharp contrast to the standard population models without selection.