Stochastic analysis on a loop group

完成了随机分析的研究，特别是对回路组的研究。由于loop群具有无穷维流形的结构，它提供了无穷维分析的一个典型例子。分析方法，例如，Dirichlet形式，由于它与空间的维数无关，在处理无穷维空间时具有优势，在loop群上定义了几类算子，研究它们的谱是一个重要的问题。关于这个问题，对数Sobolev非线性假设成立，但在固定测度下仍然是开放的。为了解决这些问题，我们发展了半群控制的一般理论，例如，Hodgy-Kodaira型拉普拉斯算子处理向量值函数，例如forms.is相当复杂。但是我们可以利用半群控制将问题简化到纯量情形。利用平方域算子给出了半群控制的一个判别准则。这个判据是用协变导数和Ricci曲率给出的，本质自伴性也可以作为半群控制的一个应用来处理。随机分析的其他问题由Shinzo Watanabe和Nobuo Yoshida研究，Watanabe在分数Sobolev空间的框架下对分布的正则性进行了改进。Yoshida建立了格点标量场的对数Sobolev不等式。进一步的相关结果包括在内。我们有几个研讨会。我们可以在许多研究人员之间交流想法和新成果。为了开展研究，这是值得的。报告了讲座摘要。

The research on stochastic analysis, in particulat on loop groups, was accomplished. Since the loop group has the structure of on infinite dimensional manifold, it offers a typical example of the infinite dimensional analysis. An analytic approach, e.g., Dirichlet forms, has an advantage to deal with infinite dimensional space because it is irrelevant of the dimension of the space.Several kinds of operators are defined on the loop group and it is an important issue to study the spectrum of the operators. Concerning this issue, the logarithmic Sobolev ineauality presumebly holds but it still remains open under the pinned measure. Aiming to solve these problems, we develop a general theory of semigroup domination for e.g., Hodgy-Kodaira type Laplacians. Dealing with vector valued function, e.g.differential forms.is rather complicate. But we can reduce the problem to the scalar case by using the semigroup domination. We gave a criterion on the semigroup domination in terms of square field operator. This criterion was given in terms of covariant derivative and Ricci curvature.We can also deal with the essential self-adjointness as an application of the semigroup domination. It can be done by combining scalar case method with the domination theorem.Other problems on stochatic analysis were studied by Shinzo Watanabe and Nobuo Yoshida.Watanabe gave a refinement of the regularity of the distribution in the framework of fractional Sobolevspace. Yoshida established a logarithmic Sobolev inequality for lattice scalar field. Further related results are included.We had several workshops. We can exchange ideas and new results among many researchers. It was worthyin order to develop the research. Summaries of lectures are presented.