Asymptotic Analysis of Probability Laws and Path Behaviors for Markov Processes

在这个研究项目中，Shimura应用蒙特卡罗方法中的技巧，考虑了从象限漂移的二维随机游动的退出概率的渐近分析。他研究了定义在马尔可夫链上的随机游动理论中的一些推论对这个问题的适用性。西冈得到了以双调和算符为生成元的双调和拟过程的“Ito演算”和“Girsanov公式”。他还研究了一类拟线性双调和方程的解的一些性质，该方程与流体力学中的一些问题有关。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动转移概率的一个渐近公式。她还考虑了一些调和映射的谱结构。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量的充分性。Ohguchi研究了一些与金融有关的马尔可夫链的数值分析，其中他考察了一些准随机数的有用性。在这个项目期间，项目成员与近藤(岛根大学)、K.Hirano(大阪大学)、T.Watanabe交换了相关的新结果。(冈山理科大学)、M.Moto(东京理工学院)、T.Yamada(立命馆大学)和S.Sato(东京电工大学)。

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.).