Asymptotic Analysis of Probability Laws and Path Behaviors for Markov Processes

在本研究项目中，Shimura采用蒙特卡罗方法，考虑了从象限漂移的二维随机行走的退出概率的渐近分析。他检验了在马尔可夫链上定义的随机游走理论的一些结果对这个问题的适用性。Nishioka得到了双调和伪过程的“Ito演算”和“Girsanov公式”，双调和算子是伪过程的产生器。他还研究了一类与流体力学有关的准线性双调和方程解的一些性质。Tanemuira精确地考虑了具有硬核势的无限布朗粒子系统的唯一性和遍历性。Kotani得到了一些无限图上随机游走转移概率的渐近公式。她还考虑了一些谐波图的光谱结构。Tsukada从算子代数的角度研究了一些随机过程和无限粒子系统的统计性的充分性。Ohguchi处理了一些与金融相关的马尔可夫链的数值分析，其中他检验了一些准随机数的有用性。在项目期间，项目成员与M.Kondo（岛根大学）、K.Hirano（大阪大学）、T.Watanabe交换了相关的近期成果。（冈山科学大学），motoo（东京工业大学），T.Yamada（立命馆大学）和S.Sato（东京电机大学）。

In this reseach project, Shimura considered asymptotic analysis for the exit probability of two dimensional random walk with drift from a quadrant, applying a technique from Monte Carlo Method. He examined applicability of some consequence from the theory of random walk defined on a Markov chain to the problem. Nishioka obtained "Ito calculus" and "Girsanov formula" for the biharmonic psudo process with the biharmnic operator as its generator. He also investigated some properties of solutions for a class of quasilinear biharmonic equations which is related with some problem from the fluid dynamics. Tanemuira considered precisely the uniqueness and the ergodicity for the infinite Brownian particle system with a hard core potential. Kotani obtained an asymptotic formula for the transition probability of a random walk on some infinite graphs. She also considered the spectral structure of some harmonic maps. Tsukada investigated the sufficiency for the statistics of some stochastic processes and infinite particle systems from the point of operator algebra. Ohguchi dealt with the numerical analysis of some Markov chains related to the finance, in which he examined usefulness of some quasi-random numbers.During the term of this project, the project members exchanged the related rescent results with M.Kondo (Shimane Univ.), K.Hirano (Osaka Univ.), T.Watanabe. (Okayama Science Univ.), M.Motoo (Tokyo Institute Technology), T.Yamada (Ritsumekan Univ.) and S.Sato (Tokyo Denki Univ.).