Stochastic analysis on a loop group

完成了随机分析，特别是环群随机分析的研究。由于环路群具有无限维流形的结构，它提供了无限维分析的一个典型例子。解析方法，例如狄利克雷形式，在处理无限维空间时具有优势，因为它与空间的维数无关。在环群上定义了几种算子，研究算子的谱是一个重要的问题。关于这个问题，对数Sobolev不等式假定成立，但它在固定测度下仍然是开放的。为了解决这些问题，我们发展了一个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，例如微分形式。相当复杂。但我们可以利用半群支配将问题简化为标量情况。给出了关于方域算子的半群控制的一个判据。这个判据是用协变导数和里奇曲率给出的。我们也可以把本质自伴随作为半群支配的一种应用来处理。可以将标量情况法与支配定理相结合。渡边晋和吉田伸夫研究了随机分析中的其他问题。Watanabe给出了分数Sobolevspace框架中分布的正则性的一个改进。吉田建立了晶格标量场的对数Sobolev不等式。包括进一步的相关结果。我们有几个工作坊。我们可以在许多研究人员之间交流思想和新成果。为了开展这项研究，这是值得的。提供讲座摘要。

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.