Evolving coalescents

在概率论和群体遗传学的交界处，联合理论已经成为一个核心的重要话题。近年来，随时间进化的聚合体结构受到了特别的关注，它们是种群谱系随时间发生变化的模型。在第一个资助期，我们把演化的合并器作为树值过程来研究，并分析了演化的Beta合并器的各种泛函的渐近分布。我们发现了一种方法，它允许将Beta聚合体的丰富的泛函表示为稳定的泊松积分，同时对于静态和演化的情况。在第二个资助期，我们将继续这方面的研究。我们将详细阐述泊松积分的方法，特别关注Beta聚合体的内部长度谱。此外，我们将放大Bolthosen-Sznitman凝聚泛函的新泛函，并将我们对凝聚泛函的研究扩展到一般的Lambda-凝聚泛函。具体地说，我们感兴趣的是外部分支的极端长度和最后一次合并的规模。我们还计划研究在个体竞争情况下家谱的对称漂移表征。

Coalescent theory has become a topic of central importance at the interface of probability theory and population genetics. In recent years, coalescent structures evolving in time have gained particular interest, they serve as a model for the changes which genealogies of populations undergo in time. In the first funding period we have studied evolving coalescents as tree-valued processes, and analysed the asymptotic distribution of various functionals of evolving Beta coalescents. We found a method which allows to represent a rich class of functionals of Beta coalescents as stable Poisson integrals, simultaneously for the static and the evolving case. In the second funding period we will continue this line of research. We will elaborate on the method of Poisson integrals, with a particular focus on the internal length spectrum of Beta coalescents. Moreover we will zoom in on new functionals of the Bolthausen-Sznitman coalescent and extend our investigations of coalescent functionals to general Lambda-coalescents. Specifically we are interested in the external branches of extremal length and the size of the last merger. We also plan to investigate the symmetric excursion representation for genealogies under individual competition.