Synthetic Research of Probability Theory

本研究由25名成员在众多概率论及相关领域研究人员的大力帮助下完成。在三年的研究期间，组织了27次会议，邀请了21名国外研究人员。在广泛的领域获得了大量的研究成果：1。讨论了概率论中的基本理论、反正弦定律在一维扩散过程中的推广、布朗运动/热核/格林函数在若干空间上的性质、马尔可夫过程和狄利克雷形式、无限维随机分析、定相法和渐近理论等。作为概率论、数学物理的应用，如相变产生的相分离面分析、自由边界问题的推导、相互作用布朗硬球的运动、渗透簇的标度极限、随机矩阵、分形上的随机过程、风险敏感随机控制问题、遍历多理论特别是ψ-混合随机变量的大数定律等。随机微分方程和非线性扩散方程的数值计算，数理统计中的渐近展开和Malliavin微积分的应用，流形坍缩等微分几何相关问题。每项研究成果的具体说明详见研究报告的小册子。如上所述，在这项研究的项目下，邀请了许多国外的研究人员，这对日本未来概率论的发展是非常富有成效的，具有极其重要的意义。

This research was accomplished by 25 members under strong helps from many researchers in probability theory and related fields. During three years of the research period, 27 meetings were organized and 21 researchers were invited from abroad. A lot of research products were obtained in broad area:1. Related to the basic theory in probability theory, extension of the inverse arcsine law to one dimensional diffusion processes, properties of Brownian motion/heat kernel/Green function on several spaces, Markov processes and Dirichlet form, infinite dimensional stochastic analysis, stationary phase method and asymptotic theory and others were discussed.2. As applications of probability theory, mathematical physics such as analysis of phase separating surface arising under phase transitions, derivation of free boundary problem, motion of interacting Brownian hard balls, scaling limit of percolation cluster, random matrix, stochastic processes on fractals, problem of risk sensitive stochastic control, ergodid theory especially law of large numbers for ψ-mixing random variables, numerical calculation for stochastic differential equations and nonlinear diffusion equations, asymptotic expansion and applications of Malliavin calculus in mathematical statistics, problems related to differential geometry like collapse of manifolds.Concrete explanations on each research result can be found in the booklet of the research report in details. As is stated above, under the project of this research, many foreign researchers were invited and it was very fruitful and extremely important for the development of the probability theory in Japan in future.