Integrated Research on Equilibrium Random Phenomena

各种重要的随机现象被认为是平衡态。在这一点上，我们已经研究了三年。在此期间，我们举办了24次专题讨论会。在这里，我们的一些结果是按照下面的关键词描述的2 .非线性与随机分析：Funaki研究了具有分形拉普拉斯的Burger方程，特别是全局解的存在性、唯一性、规律性等问题。在此基础上，他进一步研究了递进波解和自相似解白噪声分析：久保提出了一种新的白噪声空间构建方法，这种方法非常清晰和通用。他也给出了很好的表征定理。这些结果已在加尔各答印度统计研究所举行的专题讨论会上提出。Obata利用算子的符号研究了算子在空间上的表征。他把它应用到随机微分方程中信息学分析：Ihara研究了非常一般的带反馈的高斯信道，并证明了互信息的一般公式。他讨论了大偏差理论在信息论中的应用。在平稳高斯源的情况下，得到了最小误码率。Okabe根据“波动与偏差定理”的原理，将他提出的各种检验方法综合起来，形成了一个分析时间序列的综合体系。在此基础上，建立了度量向量流的KM_20-Langevin方程并加以应用自相似过程：Takenaka通过sas过程的有限维分布证明了其分布的确定性。Maejima引入了半自相似分布的概念，并将其推广到半自相似过程、算子半自相似分布等。他对它们进行了深入的研究。其他结果可以在报告和合作研究人员的论文列表中找到。少

Various important random phenomena are considered as equilibrium states. In this point of view, we have researched for three years. In the period, we held 24 symposia by the grant. Here some of our results are described along key words below.1 Stochastic Analysis : Aida got very deep results on loop spaces especially about log-Sobolev inequalities, spectral gaps, Clark-Ocone formula etc.2 Non-Linear and Stochastic Analysis : Funaki researched on Burger equations with fractal Laplacian, especially about the existence of global solution, the uniqueness, the regularities etc. Further, he researched progressive wave solutions and self-similar solutions.3 White Noise Analysis : Kubo gave a new method of construction of white noise spaces, which is very clear and general. He gave also good characterization theorems. Those results were presented in the symposium held at Indian Statistical Institute in Calcutta. Obata researched the characterization of operators on the spaces by means of their … More symbols. He applied it stochastic differential equations.4 Informatic Analysis : Ihara researched very general Gaussian channels with feed back and proved a general formula of mutual information. He discussed applications of large deviation theory to information theory. ue obtained the minimum coding rate for error in the case of stationary Gaussian source. Okabe developed an integrated system of various tests which had been proposed by him to analyze time series under the principle of "Fluctuation and Deviation Theorem". Further he constructed KM_20-Langevin equation for flow in metric vectors and applied it.5 Self Similar Process : Takenaka showed the deterministic property of distribution of SaS-processes by its finite dimensional distributions. Maejima introduced the concept of semi-self-similar distributions and extended it to that of semi-self-similar processes, operator semi-self-similar distributions and etc. He researched them very deeply.Other results are found in the report and lists of papers by co-researchers. Less