Path-properties of the tree-valued Fleming-Viot process
Path-properties of the tree-valued Fleming-Viot process
复制标题
树值 Fleming-Viot 过程的路径属性
DOI：
--
复制
发表时间：
2012
期刊：
影响因子：
--
通讯作者：
P. Pfaffelhuber
中科院分区：
文献类型：
--
作者：
A. Depperschmidt;A. Greven;P. Pfaffelhuber
本文考虑具有变异和选择的树值Fleming-Viot过程，（Xt）t ≥ 0.这个过程模拟了在无限大种群的限制下，当前存活的种群中的恢复、突变和选择下的系谱和（等位基因）类型的随机进化。系谱和类型描述的（等距类）标记度量测度空间。中性树值Fleming-Viot动力学的长时间极限是通过与金曼结合相关的标记度量测度空间给出的平衡。在本文件中，我们追求两个密切相关的目标。首先，我们证明了在固定时间t上Fleming-Viot谱系的两个著名的性质，即由对偶的性质产生的金曼结合，对整个路径都成立，这些性质与靠近其叶子的谱系树的几何有关。特别地，我们考虑在极限e → 0中个体相距不超过e的子族的数目和大小。其次，我们回答了两个开放的问题，树值Fleming-Viot过程的样本路径。我们证明了对所有的t> 0，几乎必然的标记度量测度空间Xt没有原子，并允许一个标记函数。后一个属性意味着树值Fleming-Viot过程中的所有个体都可以被唯一地分配一个类型。所有的主要结果证明了中立的情况下，然后结转到选择的情况下，通过Girsanov的公式，使绝对连续性。
We consider the tree-valued Fleming–Viot process, (Xt )t≥0 , with mutation and selection. This process models the stochastic evolution of the genealogies and (allelic) types under resampling, mutation and selection in the population currently alive in the limit of infinitely large populations. Genealogies and types are described by (isometry classes of) marked metric measure spaces. The long-time limit of the neutral tree-valued Fleming–Viot dynamics is an equilibrium given via the marked metric measure space associated with the Kingman coalescent. In the present paper we pursue two closely linked goals. First, we show that two well-known properties of the Fleming–Viot genealogies at fixed time t arising from the properties of the dual, namely the Kingman coalescent, hold for the whole path. These properties are related to the geometry of the family tree close to its leaves. In particular we consider the number and the size of subfamilies whose individuals are not further than e apart in the limit e → 0. Second, we answer two open questions about the sample paths of the tree-valued Fleming–Viot process. We show that for all t > 0 almost surely the marked metric measure space Xt has no atoms and admits a mark function. The latter property means that all individuals in the tree-valued Fleming–Viot process can uniquely be assigned a type. All main results are proven for the neutral case and then carried over to selective cases via Girsanov’s formula giving absolute continuity.
影响因子：
2
作者：
A. Greven;P. Pfaffelhuber;A. Winter
通讯作者：
A. Winter