1995 Fiscal Year Final Research Report Summary
Research on differential operators in infinite dimensional spaces with symmetry
| Project/Area Number |
06452014
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | The University of Tokyo |
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Principal Investigator |
KUSUOKA Shigeo Professor, Graduate School of Mathematical Sciences, The University of Tokyo, 大学院・数理科学研究科, 教授 (00114463)
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| Co-Investigator(Kenkyū-buntansha) |
TSUTSUMI Yoshio Assistant Professor, Graduate School of Mathematical Sciences, The University of, 大学院・数理科学研究科, 助教授 (10180027)
OSADA Hirofumi Assistant Professor, Graduate School of Mathematical Sciences, The University of, 大学院・数理科学研究科, 助教授 (20177207)
KATAOKA Kiyoomi Professor, Graduate School of Mathematical Sciences, The University of Tokyo, 大学院・数理科学研究科, 教授 (60107688)
MATSUMOTO Yukio Professor, Graduate School of Mathematical Sciences, The University of Tokyo, 大学院・数理科学研究科, 教授 (20011637)
MATANO Hiroshi Professor, Graduate School of Mathematical Sciences, The University of Tokyo, 大学院・数理科学研究科, 教授 (40126165)
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| Project Period (FY) |
1994 – 1995
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| Keywords | Wiener Measure / Stochastic Analysis / Skelton / Dolbealt Cohomology / Horomorphic Function / Infinite Dimensional Analysis / Complex Wiener Space |
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| Research Abstract |
The object of this research was differential operators in infinite dimensional spaces, especially ones with symmetry. We had a progess in the research on holomorphic functions on complex Wiener spaces from viewpoint of stochastic analysis and in the research on asymptotics of fundamental solutions of heat equations with boundary conditions.
There have been some works on holomorphic functions in complex Wiener spaces. But all of them handled holomorphic functions defined in the whole space only. So these works did not make clear the holomorphicity as a local property. Since the aim of this research is to establish analysis on Wiener manifolds, we really need the localization of the concepts. We had a big progress in the localization of the notion of holomorphicity and skelton. The skelton of holomorphic functions is holomorphic functions in the Cameron-Martin space associated the original holomorphic functions. The existence of the skelton for holomorphic functions defined on the whole c
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omplex Wiener space was shown by Prof.Sugita in Kyushu University. But nothing is known about the skelton for holomorphic functions defined in subdomains. In this research we showed that if the subdomain is given by a positive domain of a good function, then there exists the skelton of holomorphic functions defined in the domain and the corespondance of skelton and holomorphic functions is one-to-one. Combining this result with known results, we can see that the Dolbeault cohomology coincides with the Cech cohomology of the sheaf of holomorphic functions in the domain. We hope that this result plays an important role in complex analysis in infinite dimensional spaces.
We also obtained the precise estimates on the asymptotic behavior of the fundamental solution for the heat equation with Dirichlet or Neumann conditions in the outside of strictly convex domains with smooth boundaries. We obtained this result by representing the fundamental solutions by the Wiener measure and by using the symmetry. We did not expect such a result, but we obtained it as a by-product. Less
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