Genealogy of Fleming-Viot processes

首席调查员调查了一个垫脚石弗莱明-维奥特过程的谱系问题。他得到了以下结果：平均概率，两个基因随机选择在稳态的扩散有一个共同的祖先是书面长期的合并时间的双重过程。假设菌落的集合是d维格子，并且迁移过程是随机游走。然后，为了使两个基因具有概率为1的共同祖先，维度d小于或等于2是必要且充分的。从共同突变开始的所有突变数等于s的平均概率由泊松分布的混合分布给出。接下来，首席研究员研究了具有无限多个等位基因的垫脚石模型的有限基因样本中的等位基因数。他的动机是试图推广著名的结果的情况下，没有地理结构的垫脚石模型。这是与T.Soshi的联合工作。研究者N.Konno研究了无限粒子系统。他在数学方面和应用领域取得了许多成果。宫原洋三研究了与不完全资产市场相关的典型鞅测度。研究者T.Misawa研究了随机动力系统。他通过观察动力系统的对称性得到了守恒量。

The head investigator has investigated a genealogical problem of a stepping stone Fleming-Viot process. He got the following results : The average probability that two genes randomly chosen in the stationary state of the diffusion have a common ancestor is written in the term of the coalescent time of the dual process. Assume that the set of colonies is the d-dimensional lattice, and that the migration process is a random walk. Then, in order that the two genes have a common ancestor with probability 1, it is necessary and sufficient that the dimension d is less than or equal to 2. The average probability that the number of all mutations since the common anscestor equals s is given by a mixture of Poisson distributions.Next, the head investigator has investigated the number of alleles in a sample of finite genes for stepping stone models with infinitely many alleles. His motivation is to try to extend the well-known results in the case without geographical structure for stepping stone models. This is a joint work with T.Soshi.The investigator, N.Konno, has investigated infinite particle systems. He got many results in mathematical aspects and applied fields. The investigator, Y.Miyahara has investigated canonical martingale measures related to incomplete assets markets. The investigator, T.Misawa, has investigated stochastic dynamical systems. He got conserved quantities observing symmetry of dynamical systems.