Genealogy of Fleming-Viot processes

首席调查员调查了垫脚石弗莱明-维奥特过程的系谱问题。他得到了如下结果：在扩散的静止状态中随机选择的两个基因具有共同祖先的平均概率被写成对偶过程的合并时间。假设群体的集合是d维格子，迁移过程是随机游走。然后，为了使这两个基因有一个概率为1的共同祖先，维度d小于或等于2是充要条件。自共同变异者等于S以来，所有突变个数的平均概率是由泊松分布的混合给出的。接下来，首席研究员研究了具有无限多个等位基因的踏脚石模型的有限个基因样本中的等位基因数目。他的动机是试图在没有地理结构的情况下推广众所周知的结果作为垫脚石模型。这是与T.Soshi.合作的工作。研究员N.Konno研究了无限大的粒子系统。他在数学方面和应用领域都取得了许多成果。调查员Y.Miyahara研究了与不完全资产市场相关的规范鞅度量。研究者T.Misawa研究了随机动力系统。观察动力系统的对称性，他得到了守恒量。

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.