1997 Fiscal Year Final Research Report Summary
Basic Studies on Analysis of Stochastic Variation.
| Project/Area Number |
07640280
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Saitama University |
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Principal Investigator |
DOKU Isamu Saitama Univ.Faculty of Education Assoc.Prof., 教育学部, 助教授 (60207686)
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| Co-Investigator(Kenkyū-buntansha) |
KIMURA Takashi Saitama Univ.Faculty of Education Assoc.Prof., 教育学部, 助教授 (00195364)
TAKIJIMA Kunio Saitama Univ.Faculty of Education Prof., 教育学部, 教授 (30015812)
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| Project Period (FY) |
1995 – 1997
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| Keywords | white noise analysis / stochastic variation / infinite dimensional analysis / Gaussian generalized functionals / infinite dimensional Fourier type transform / stochastic boundary value problem |
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| Research Abstract |
We construct on an infinite dimensional manifold a new type of Laplacian DELTA_p associated with the Hida differential in white noise analysis. This Laplacian is completely different from the well known Hida Laplacian DELTA_H. Our Laplacian is realized so nicely as an operator having C^* invariance that we can show an infinite dimensional version of the De Rham-Hodge-Kodaira decomposition theorem. We define a Pseudo-Fourier-Mehler transfrom PSI (PFM transform for short) and derive several fundamental properties. Actually the PFM transform is a variant of infinite dimensional Fourier type transform in white noise analysis. Intertwining properties and Fock expansion of the PFM transfrom are derived. Moreover, we can show that a family of PFM transforms forms a regular one-parameter subgroup of the linear homeomorphism group, and the corresponding infinitesimal generator is determined. In addition, we can extend it to a more generalized transform and the characterization theorem is proved. We consider the stochastic variational equation and find the solution CHI (s) lying in the white noise space, and an explicit representation of deltaCHI is also obtained. Finally, we consider the stochastic boundary value problem under the general setting of white noise analysis. Based on the formulation of stochastic asymptotic solutions, we can construct the solutions. A limit theorem for such solutions is derived from the point of view of oscillatory discussion for the random system.
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