| Project/Area Number |
15340039
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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 |
Basic analysis
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| Research Institution | Tohoku University |
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Principal Investigator |
OBATA Nobuaki Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (10169360)
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| Co-Investigator(Kenkyū-buntansha) |
HIAI Fumio Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (30092571)
URAKAWA Hajime Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (50022679)
HORA Akihito Okayama University, Graduate School of Natural Science, Associate Professor, 大学院・自然科学研究科, 助教授 (10212200)
ARIMITSU Toshihico University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor, 大学院・数理物質科学研究科, 教授 (50134200)
SAITO Kimiaki Meijo University, Faculty of Science and Technology, Professor, 理工学部, 教授 (90195983)
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| Project Period (FY) |
2003 – 2005
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| Project Status |
Completed (Fiscal Year 2005)
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| Budget Amount *help |
¥15,500,000 (Direct Cost: ¥15,500,000)
Fiscal Year 2005: ¥5,200,000 (Direct Cost: ¥5,200,000)
Fiscal Year 2004: ¥4,800,000 (Direct Cost: ¥4,800,000)
Fiscal Year 2003: ¥5,500,000 (Direct Cost: ¥5,500,000)
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| Keywords | White noise theory / Quantum probability / Quantum white noise / Quantum decomposition / Quantum central limit theorem / Levy Laplacian / Dissipative quantum system / Spectral analysis of graphs |
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| Research Abstract |
Stochastic analysis has developed considerably keeping profound applications to mathematical analysis of random phenomena, where the traditional framework, called the Ito calculus, is based upon functionals of Brownian motion. We start, in this research project, with the new aspect of quantum white noise which is obtained by applying quantum decomposition and time derivative to the Brownian motion. Since the quantum white noise is more primary, we come naturally to two lines of research beyond the traditional stochastic analysis : nonlinear theory and non-Gaussian extension. We developed quantum white noise theory by making research collaboration, workshops, summer schools, international conferences. The research subjects are as follows :
(1) Nonlinear white noise equations : We developed white noise operator theory by means of a newly obtained characterization theorem and introduced the notion of quantum white noise derivatives.
(2) Complex white noise : Unitarity of the Fourier-Gauss t
… More
ransform is proved. We found that the solution to normal-ordered white noise equation involving quadratic quantum white noise possesses partial unitarity.
(3) Levy Laplacian and higher powers of quantum white noises : Associated stochastic processes are constructed and the relation between them is established through the heat equation.
(4) Interacting Fock space : We established a systematic approach to asymptotic spectral analysis of graphs, in particular, for growing regular graphs and graphs related to notions of independence.
(5) Non-Gaussian stochastic processes : We are continuing the study of interacting Fock space of multi-mode towards quantum decomposition of stochastic processes.
(6) Infinite dimensional harmonic analysis : The method of quantum decomposition is applied to prove a generalization of the Kerov central limit theorem. The asymptotic representation theory of the symmetric groups and the infinite wreath products have been developed.
(7) Application to quantum physics : We promoted collaboration with physicists through quantum stochastic differential equations, statistical properties of turbulence, Bose-Einstein condensation on graphs. Less
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