Studies on Probability theory.

本课程涉及随机过程理论及相关领域。为了促进这些领域的活动，我们在1987年4月至1989年3月期间举办了各种专题的讲习班和研讨会。在该项目支持的会议上，我们对下述专题的研究感兴趣：狄利克雷空间和马尔可夫过程，Malliavin演算及其应用，高斯过程，平稳过程和时间序列，Langivin方程，随机过程的极限定理，分形上的随机过程，随机介质中的随机过程，随机介质中的谱波动，渗流，Ising模型，相变和临界现象，遍历理论，分析和几何上的循环空间，随机控制，大偏差和应用，应用非标准分析的概率论等像许多其他分支的数学，概率论已经成长和发展的灵感来自其他领域的科学。这一点在随机过程理论中尤为明显，例如，数学物理和工程学的思想对随机过程理论产生了显著的影响。在一系列的讲习班举办的内山和船木，一个认真的尝试已经取得了概率方法在数学物理。特别地，我们致力于以下主题的研究：时间相关的Ginzburg-Landau模型和相变，具有许多障碍物的区域中的Laplacian，各种模型的流体动力学极限。随机分析是概率论的一个新的分支，它与偏微分方程、几何、遍历理论、数学物理等学科有着密切的联系，正变得越来越重要。为了促进随机分析的活动，我们还组织了一系列研讨会。在这些会议上，宣布了与随机分析中的各种主题有关的显着结果。

We have concerned with the theory of stochastic processes and related fields. For the purpose of promoting the activity in these fields, we held workshops and symposia on various topics during the period between 1987, April and 1989, March. In the meetings supported by the project, we were interested in the studies of the topics described below: Dirichlet space and Markov process, Malliavin calculus and applications, Gaussian process, stationary process and time series, Langivin equation, limit theorem for stochastic processes, stochastic process on fractals, stochastic process in random media, fluctuation of spectra in random media, percolation, Ising model, phase transition and critical phenomena, ergodic theory, analysis and geometry on loop space, stochastic control, large deviation and applications, applications of non-standard analysis to probability theory etc. Like many other branches of mathematics, probability theory has grown and developed with inspiration from other areas of science. This is especially clear in the theory of stochastic processes, where ideas from mathematical physics and engineering, for example, have exerted a noteworthy influence. In a series of the workshops organized by Uchiyama and Funaki, a serious attempt has been made to probabilistic methods in mathematical physics. In particular, we were devoted to the sutdy of the following topics: time dependent Ginzburg-Landau model and phase transitions, the laplacian in regions with many obstacles, hydrodynamic limit of various models. Stochastic analysis is a new branch of probability theory and is becoming more and more important in close connection with partial differential equations, geometry, ergodic theory and mathematical physics. For purpose of promoting the activity in stochastic analysis we have also organized a series of workshops. In these meetings, notable results related to various topics in stochastic analysis were announced.