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1997 Fiscal Year Final Research Report Summary

Basic Studies on Analysis of Stochastic Variation.

Research Project

Project/Area Number 07640280
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionSaitama University

Principal Investigator

DOKU Isamu  Saitama Univ.Faculty of Education Assoc.Prof., 教育学部, 助教授 (60207686)

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)
Project Period (FY) 1995 – 1997
Keywordswhite noise analysis / stochastic variation / infinite dimensional analysis / Gaussian generalized functionals / infinite dimensional Fourier type transform / stochastic boundary value problem
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.

  • Research Products

    (15 results)

All Other

All Publications (15 results)

  • [Publications] I.Doku: "Pseudo-Fourier-Mehler transform in white noise analysis and application to lifted convergence to a certain approximation Cnudly problem." J.Saitama Univ.Math.Nut.Sci.II. 44-1. 55-79 (1995)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] I.Doku: "On the Laplacian on a space of white noise functionals." Tsukuba J.Math.19. 93-119 (1995)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] I.Doku: "On Psendo-Fourier-Mehler transforms and infinitesimal generators in white noise calowas." RIMS Kokyuroku(Kyoto Univ). 923. 33-50 (1995)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] I.Doku: "On a class of infinite dimensional Fourier type transforms in white noise calculus." Proceedings of the Seventh Japan-Russia Symp.Probab Theory Math.Stat.(world scentific). JRPS95. 51-61 (1996)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] I.Doku: "A Fock expansion of the Pseudo-Fourier-Mehler transform." J.Saitama Univ.Math.Nat,Sci.45-1. 11-16 (1996)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] I.Doku: "White noise analysis and the boundary value problem in the space of stochastic distributions." RIMS Kokyuroku(Kyoto Univ). 957. 36-53 (1996)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] I.Doku: "Pseudo-Fourier-Mehler transform in white noise analysis and application of lifted convergence to a certain approximation Cauchy problem" J.Saitama Univ.Math.Nat.Sci.II.44-11. 55-79 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] I.Doku: "On the Laplacian on a space of white noise functionals" Tsukuba J.Math.19. 93-119 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] I.Doku: "Mathematical aspect of complex canonical quantization in quantum gravity" J.Saitama Univ.Math.Nat.Sci.44-2. 1-11 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] I.Doku: "Tomonaga-Schwinger supermany-time formalism and Hida's stochastic calculus" J.Saitama Univ.Math.Nat.Sci.44-2. 9-19 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] I.Doku: "On Pseudo-Fourier-Mehler transforms and infinitesimal generators in white noise calculus, in Colloquium on Analysis of Operators on Gaussian Space and Quantum Probability Theory" RIMS Kokyuroku (Kyoto Univ.). 923. 33-51 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] I.Doku: "On a class of infinite dimensional Fourier type transforms in white noise calculus" Proceedings of the Seventh Japan-Russia Symposium on Probability Theory and Mathematical Staatistics, JRPS95, Tokyo, July.26-30 1995, World Scientific, Singapore. 51-61 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] I.Doku: "Some intertwining properties of the Pseudo-Fourier-Mehler transform" J.Saitama Univ.Math.Nat.Sci.45. 1-9 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] I.Doku: "A fock expansion of the Pseudo-Fourier-Mehler transform" J.Saitama Univ.Math.Nat.Sci.45-1. 11-16 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] I.Doku: "White noise analysis and the boundary value problem in the space of stochastic distributions, in Colloquium on Quantum Stochastic Analysis and Related Fields" RIMS Kokyuroku (kyoto Univ.). 957. 36-53 (1996)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 1999-03-16  

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