Co-operative Research of Probability Theory

这项合作研究的目的是在理论和应用随机分析的系统发展。为此，首席研究员和研究员在两年内组织了四个不同类别的14次研讨会：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：对概率分布和随机场的具体研究每次研讨会都包括对最近进展的一般性评论、对新结果的报告和对未来发展的讨论。本研究在每个类别中的具体成果是：a）Malliavin演算和Dirichlet形式理论的发展b）分形集上扩散过程和谱分析的进展c）量子混沌，控制论，信息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认识我们举办了两次大型会议，其中150多名数学家参加了上述四个类别的混合，以创造对随机分析在理论和应用方面的深度和广度的新认识及其广阔的未来。

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.