Particle configurations for branching Brownian motion with an inhomogeneous branching rate

IF 0.6 4区 数学 Q4 STATISTICS & PROBABILITY Alea-Latin American Journal of Probability and Mathematical Statistics Pub Date : 2021-11-30 DOI:10.30757/alea.v20-28
Jiaqi Liu, Jason Schweinsberg
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引用次数: 3

Abstract

Aiming to understand the distribution of fitness levels of individuals in a large population undergoing selection, we study the particle configurations of branching Brownian motion where each particle independently moves as Brownian motion with negative drift, particles can die or undergo dyadic fission, and the difference between the birth rate and the death rate is proportional to the particle's location. Under some assumptions, we obtain the limit in probability of the number of particles in any given interval and an explicit formula for the asymptotic empirical density of the fitness distribution. We show that after a sufficiently long time, the fitness distribution from the lowest to the highest fitness levels approximately evolves as a traveling wave with a profile which is asymptotically related the the Airy function. Our work complements the results in Roberts and Schweinsberg (2021), giving a fuller picture of the fitness distribution.
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具有非均匀分支率的分支布朗运动的粒子构型
为了了解个体在大规模选择群体中的适应度分布,我们研究了分支布朗运动的粒子构型,其中每个粒子独立地作为具有负漂移的布朗运动运动运动,粒子可以死亡或发生并进裂变,出生率和死亡率之间的差异与粒子的位置成比例。在某些假设下,我们得到了在任何给定区间内粒子数的概率极限,并给出了适应度分布的渐近经验密度的显式公式。我们证明,在足够长的时间后,从最低适应度到最高适应度的适应度分布近似演化为行波,其轮廓与Airy函数渐近相关。我们的工作补充了Roberts和Schweinsberg(2021)的结果,更全面地描述了适合度分布。
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来源期刊
CiteScore
1.10
自引率
0.00%
发文量
48
期刊介绍: ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted. ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper. ALEA is affiliated with the Institute of Mathematical Statistics.
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