The spatial Lambda-Fleming-Viot process: an event-based construction and a lookdown representation
The spatial Lambda-Fleming-Viot process: an event-based construction and a lookdown representation
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空间 Lambda-Fleming-Viot 过程：基于事件的构造和下视表示
DOI：
10.1214/13-aihp571
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
A. Wakolbinger
中科院分区：
文献类型：
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作者：
A. Véber;A. Wakolbinger
我们构造了一个与空间Lambda-Fleming-Viot过程(SLFV)等价的测值等价过程。与[eth08]中的构建相比，我们固定了繁殖事件序列的实现，并获得了局部遗传多样性的猝灭进化。为此，我们使用了一种粒子表示法，它突出了谱系在将类型(或等位基因)归因于群体中的个体方面的作用。这种构造还使我们能够澄清SLFV的状态空间，并推导出度量值过程以及描述个体样本之间谱系关系的标记树的几个路径属性。我们用一个俯视结构来补充它，它提供了一个粒子系统，它在时间t的经验分布被视为t中的一个过程，具有熄灭的SLFV定律。在所有这些结果中，我们使用固定的事件配置以及只在空间中局部发生复制的事实带来了严重的技术问题，这些问题可以通过控制宏观时间间隔内特定区域发生的事件和存在的粒子的数量来克服。
We construct a measure-valued equivalent to the spatial Lambda-Fleming-Viot process (SLFV) introduced in [Eth08]. In contrast with the construction carried out in [Eth08], we fix the realization of the sequence of reproduction events and obtain a quenched evolution of the local genetic diversities. To this end, we use a particle representation which highlights the role of the genealogies in the attribution of types (or alleles) to the individuals of the population. This construction also enables us to clarify the state-space of the SLFV and to derive several path properties of the measure-valued process as well as of the labeled trees describing the genealogical relations between a sample of individuals. We complement it with a look-down construction which provides a particle system whose empirical distribution at time t, seen as a process in t, has the law of the quenched SLFV. In all these results, the facts that we work with a fixed configuration of events and that reproduction occurs only locally in space introduce serious technical issues that are overcome by controlling the number of events occurring and of particles present in a given area over macroscopic time intervals.