Fleming-Viot Processes with Geographical Structures

(1)考虑状态空间为可数的正递马氏链的第一通过时的分数阶矩。利用转移概率对平稳分布的收敛速度，得到了分数阶矩的有限判据。作为应用，证明了直积过程的通过时间与单马氏链的通过时间具有相同的分数阶矩。[1](2)研究了具有无限多个等位基因的踏脚石模型。计算了模型定态下有限粒子样本中不同元素的平均数，并讨论了该量的强迁移极限。(3)讨论了由归一化从属项导出的随机离散概率分布。得到了一个推广的Ewens抽样公式。文[2](4)讨论了随机Logistic模型的平均灭绝时间，得到了它的渐近行为。(5)讨论了FAA di Bruno公式。给出了该公式的一个初等证明，并讨论了它的应用。[3](6)Miyahara得到了许多关于几何Levy过程和极小鞅测度的结果。他还讨论了它在数学金融中的应用。[4]，[5]，[6]，[7](7)三泽在随机微分方程的数值分析中得到了许多结果。他提出了一种新的分析方法。以这种方式，随机微分方程的近似解保持了SDE的守恒量。

(1) Fractional moments of the first passage-times are considered for positively recurrent Markov chains with countable state space. A criterion of the finiteness of the fractional moments is obtained in terms of convergence rate of transition probability to the stationary distribution. As an application it is proved that the passage time of a direct product process has the same order of the fractional moments as that of the single Markov chain.[1](2) The stepping stone model with infinite many alleles is studied. The average number of distinct elements in a sample of finite particles in the stationary state of the model is calculated, and the strong-migration-limit of the quantity is discussed.(preprint)(3) Random discrete probability distributions derived from normalized subordinators are discussed. A generalized Ewens' sampling formula is obtained. The asymptotical behavior of the average length of Young diagrams is made clear in terms of Levy measure.[2](4) The average extinction time for a stochastic logistic model is discussed, and its asymptotic behavior is obtaind.(preprint)(5) The formula of Faa di Bruno is discussed. An elemntary proof of the formila is obtained, and its applications are discussed.[3](6) Miyahara obtained many results on geometric Levy processes and minimal martingale measures. He also disccussed its applications to mathernatical finance.[4], [5], [6], [7](7) Misawa obtained many results on numerical analysis on stochastic differential equations. He proposed a new method on the analysis. Approximate solutions, in this manner, to a stochastic differential equation preserve conserved quantities of the SDE.