1997 Fiscal Year Final Research Report Summary
Stochastic analysis on a loop group
| Project/Area Number |
08454041
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
SHIGEKAWA Ichiro Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (00127234)
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| Co-Investigator(Kenkyū-buntansha) |
YOSHIDA Nobuo Kyoto Univ., Graduate School of Science, Assistant, 大学院・理学研究科, 助手 (40240303)
NOMURA Takaaki Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (30135511)
TANIGUCHI Masahiko Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (50108974)
IWASAKI Nobuhisa Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70027374)
WATANABE Shinzo Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90025297)
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| Project Period (FY) |
1996 – 1997
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| Keywords | loop group / stocastic differential equation / inifinite dimensional analysis / essential self-adjointness / Ricci curvature / spectrum / semigroup domination / logarithmic Sobolev inequality |
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| Research Abstract |
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.
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