Co-operative Research of Probability Theory

这次合作研究的目的是在理论和应用上系统地发展随机分析。为此，首席研究员和研究人员在两年内组织了四个不同类别的14个研讨会：a)数学中的随机分析：无限维分析、有限维分析和马尔可夫过程b)基于随机分析的数学分形集c)随机分析在其他领域的应用：数学物理、工程科学和经济学d)概率论中的随机分析：关于概率分布和随机场的具体研究每个研讨会都包括对最近进展的概述、新结果的报告和对未来发展的讨论。这项研究在每个类别中的具体成果是：a)Malliavin微积分和Dirichlet形式理论的发展b)扩散过程和分形集的谱分析的进展c)量子混沌、控制理论、信息理论和金融理论的数学进展d)对单峰分布和稳定随机场的新理解我们有两个大型会议，150多名数学家参加了会议，以上四个类别混合在一起，对随机分析的理论和应用的深度和广度以及它的前景有了新的理解。

This co-operative research aims at a systematic development of stochastic analysis both in theory and applications. To this end, the chief investigator and investigators organized 14 symposiums of four different categories in two years:a)Stochastic analysis in mathematics: infinite dimensional analysis, finite dimensional analysis and Markov processesb)Mathematics fractal sets based on stochastic analysisc)Applications of stochastic analysis to other fields: Mathematical physics, engineering science and economicsd)Stochastic analysis within probability theory: specific investigations on probability distributions and random fieldsEach symposium contained general reviews on recent progress, reports on new results and discussions for future developments. Specific achievements by this research in each category are:a)development in Malliavin calculus and theory of Dirichlet formsb)progress on diffusion processes and spectral analysis on fractal setsc)mathematical progress on quantum chaos, control theory, information theory and finance theoryd)new understanding on unimodal distributions and stable random fieldsWe had two big conferences in which more than 150 mathematicians participated and the above four categories are mixed together to create a new understanding on the depth and width of stochastic analysis both in theory and applications and on its promising future.