2004 Fiscal Year Final Research Report Summary
Integrated research of Probability Theory
| Project/Area Number |
14204008
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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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 University, Graduate School of Science, Department of mathematics, Professor, 大学院・理学研究科, 教授 (00127234)
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| Co-Investigator(Kenkyū-buntansha) |
KUMAGAI Takashi Kyoto University, Research Institute of Mathematical Science, Associate Professor, 数理解析研究所, 助教授 (90234509)
HINO Masanori Kyoto University, Graduate School of Informatics, Associate Professor, 大学院・情報学研究科, 助教授 (40303888)
AIDA Shigeki Osaka University, Graduate School of Fundamental Engineering, Professor, 大学院・基礎工学研究科, 教授 (90222455)
OGURA Yukio Saga University, Faculty of Science, Professor, 理工学部, 教授 (00037847)
SHIRAI Tomoyuki Kyushu University, Graduate School of Science, Associate Professor, 大学院・理学研究院, 助教授 (70302932)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Stochastic analysis / Brownian motion / Wiener space / logarithmic Sobolev inequality / spectral gap / Schrodinger operator / Littlewood-Paley inequality / Riemannian manifold |
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| Research Abstract |
The main research area of the head investigator is diffusions in infinite dimension. At the same time, a research of diffusions on a Riemannian manifold is accomplished because geometric point of view is important in our research. We gave a probabilistic proof of the Littlewood-Paley inequality, the L^P norm equivalence between the gradient and the square of the generator, essential self-adjointness of a Schrodinger operator, and the spectral gap. The essential matter we used is the intertwining property between gradient and the generator. The use of Functional Analysis is crucial because it is irrelevant of the geometry of the space. In our general framework, we assume the logarithmic Sobolev inequality and the exponential integrability of the remaining term of the intertwining property. We also discussed the similar problem in a setting of Riemannian manifold with convex boundary.
Further we considered a Schrodinger operator on the Wiener space of the form L+V,L beging the Ornstein-Uh
… More
lenbeck operator. We gave a characterization of the generator domain and proved the essential self-adjointness and the spectral gap of the Schrodinger operator under a suitable condition of the potiential V.
We also showed the Littlewood-Paley inequality for the the Schrodinger operator. Since we have a potential term, we need a modification of the standard proof. This method works for a Hodge-Kodaira operator on a Riemannian manifold with a potential.
In this project, we have held several symposiums and gave financial support for participants. One of them is "Stochastic Analysis and related fields" that was a research project of the Research Institute of Mathematical Sciences in 2002. We invited Professors McKean, Ustunel, Rockner, etc., from abroad. The others are "Probability Summer School" in 2003,2004. We gave introductory lectures of frontier of recent research for graduated students. We could accomplish stimulating discussions. We also held every year "Stochastic Processes and related fields", which gave fruitful communication between researchers in Japan. As a sum, we held 24 symposiums during three years and invited 14 foreign researchers and produced many results. Less
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