2003 Fiscal Year Final Research Report Summary
Study on global properties of local martingales and martingales on manifolds.
| Project/Area Number |
13640170
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Keio University |
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Principal Investigator |
ATSUJI Atsushi Keio Univ., Fac. Economics, Professor, 経済学部, 教授 (00221044)
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| Co-Investigator(Kenkyū-buntansha) |
KOTANI Shinichi Osaka Univ., Grad. School Science, Professor, 大学院・理学研究科, 教授 (10025463)
SUZUKI Yuki Keio Univ., Fac. Medicine, Lecturer, 医学部, 講師 (30286645)
TOMURA Yozo Keio Univ., Fac. Science and Technology, Associate Prof., 理工学部, 助教授 (50171905)
TAKEGOSHI Kensho Osaka Univ., Grad. School Science, Associate Prof., 大学院・理学研究科, 助教授 (20188171)
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| Project Period (FY) |
2001 – 2003
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| Keywords | Brownian motion / local martingale / T-Martingale / δ-subharmonic function / Liouville theorem / missional surface / Nevanlinna theory / subharmonic functions |
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| Research Abstract |
Head investigator A.Atsuji obtained the following main three results.
1. For a minimal surface which may not be properly immersed, its total curvature is finite if its projective volume is finite. In this result we used some properties of Brownian motion on the surfaces which is martingale on the ambient spaces. Lemma for logarithmic derivative of δ-subharmonic functions which is proved by using properties of 1-dimensional local martingale works well.
2. Parabolicity is characterized by δ-subharmonic functions. Using this characterization we showed some Liouville type theorems for harmonic maps of finite energy from parabolic manifolds to Hadamard manifolds. We used some properties of martingales given as images of Brownian motion by harmonic maps.
3. A Nevanlinna theory for meromorphic functions on complex submanifolds in C^n is obtained. S.Kotani gave a necessarily and sufficient condition for 1 dimensional diffusions to be true martingales. K.Takegoshi gave some criteria for complete Riemannian manifolds to be parabolic, Liouville type theorems for harmonic maps and some results on isometry of conformal metric of positive scalar curvature. Y.Suzuki same results on long time behavior of diffusion processes in random environment with one-sided Brownian potential.
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