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Stochastic Analysis and Asymptotics

Research Project

Project/Area Number 08454046
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

IKEDA Nobuyuki  Ritsumeikan Univ., Faculty of Science and Engineering, Professor, 理工学部, 教授 (00028078)

Co-Investigator(Kenkyū-buntansha) TAKAYAMA Yukihide  Ritsumeikan Univ., Faculty of Science and Engineering, Associate Professor, 理工学部, 助教授 (20247810)
NATSUME Toshikazu  Ritsumeikan Univ., Faculty of Science and Engineering, Professor, 理工学部, 教授 (00125890)
SHI'NYA Hitoshi  Ritsumeikan Univ., Faculty of Science and Engineering, Professor, 理工学部, 教授 (70036416)
ARAI Masharu  Ritsumeikan Univ., Faculty of Science and Engineering, Professor, 理工学部, 教授 (20066715)
YAMADA Toshio  Ritsumeikan Univ., Faculty of Science and Engineering, Professor, 理工学部, 教授 (10037749)
Project Period (FY) 1996
Project Status Completed (Fiscal Year 1996)
Budget Amount *help
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 1996: ¥3,200,000 (Direct Cost: ¥3,200,000)
Keywordsstochastic analysis / Wiener integral / path integral / Van Vleck formula / Jacobi field / local time / Selberg trance formula / Wiener測度
Research Abstract

1. The Van Vleck formula for Feynman's path integrals with qusdratic phase functionals plays an important role in the study of the mathematical science. By using the method of stochastic analysis, N.Ikeda proved an analogous formula for Weiner integrals with quadratic phase functionals to the Van Vleck formula mentioned above. By using transformations described in terms of Jacpbi fields, we can rewrite quadratic Wiener functionals associated with quadratic Lagrangians in the canonical form on the Wiener space. Then the functional determinants appeared in the Van Vleck formula can be regarded as the Jacobian of the transformation above as in case of finite dimensional manifolds.
2. Without using the exact formula of the fundamental solution of the heat equation on the Poincare upper half plane, N.Ikeda obtained a proof of Selberg trace formula based on the integral representation of the fundamental solution of the heat equation on the Wiener space.
3. T.Yamada has studied principal values of Brownian local times and representations of Brownian additive functionals of zero energy. M.Arai obtained remarkable results about the growth order of eigenfuctions of Schrodinger operators with potentials admitting some integral conditions.
T.Natsume has studied several index theorems by using the method of functional analysis.
Y.Takayama and Y.Sato have studied several problems of mathematical science from the point of view of the computer science.

Report

(2 results)
  • 1996 Annual Research Report   Final Research Report Summary
  • Research Products

    (22 results)

All Other

All Publications (22 results)

  • [Publications] N. Ikeda, S. Manabe: "Van Vleck-Pauli formula for Wiener integrals and Jacobi fields" Ito's stochastic calculus and probability theory. Springer. 141-156 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] N. Ikeda, S.Kusuoka: "Shot time asymptotics for fundamental solutions of heat equations with boundary conditions" Proceedings of a Taniguchi International Workshop, World Scientific. 予定. (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T. Yamada: "Principal values of Brownian local times and their related topics" Itos stochastic calculus and probability theory. Springer. 413-422 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] M. Arai, J. Uchiyama: "Growth order of eigenfunctions of Schrodinger operators with potentials admitting some integral conditions I-General theory-" Publ. R. I. M. S. Kyoto Univ.32. 581-616 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] M. Arai, J. Uchiyama: "Growth order of eigenfunctions of Schrodinger operators with potentials admiting some integral conditions II-Applications-" Publ. R. I. M. S. Kyoto Univ.32. 617-637 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] G. A. Elliott, T. Natsume, R. Nest: "The Atiya-Singer index theoremas passage to the classical limit on quantum mechanics" Comm. Math. Phys.182. 505-533 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] H. Moriyoshi, T. Natsume: "The Godbillon-Vey cocycle and logitudinal Dirac operators" Pacific J. Math.172. 483-539 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Y. Takayama: "Extraction of concurrent processes from higher dimensional automata" Lecture Notes in Computer Science, Springer. 1059. 72-86 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] N.Ikeda, S.Manabe: "Van Vleck-Pauli formula for Wiener integrals and Jacobi fields" Ito's stochastic calculus and probability theory ed.by N.Ikeda et al, Springer. 141-156 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] N.Ikeda, S.Kusuoka: "Short time asymptotics for fundamental solutions of heat equations with boundary conditions" Proceedings of a Taniguchi International Workshop.ed.by.K.D.Elworthy et al., World Scientific.(to appear). (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Yamada: "Principal values of Brownian local times and their related topics" Ito's stochastic calculus and prohability theory, ed.by N.Ikeda et al., Springer. 413-422 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] M.Arai, J.Uchiyama: "Growth order of eigengunctions of Schrodinger operators with potentials admiting some integral conditions I--General theory" Publ.R.I.M.S.Kyoto Univ.32. 581-616 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] M.Arai, J.Uchiyama: "Growth order of eigenfunctions of Schrodinger operators with potentials admiting some integral conditions II--Applications" Publ.R.I.M.S.Kyoto Univ.32. 617-637 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] G.A.Elliott, R.T.Natsume, R.Nest: "The Atiya-Singerindex theoremas passage to the classical limit on quantum mechanics" Comm.Math.Phys.182. 505-533 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] H.Moriyoshi, T.Natsume: "The Godbillon-Vey cocycle and logitudinal Dirac operators" Pacific J.Math. 172. 483-539 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] Y.Takayama: "Extraction of concurrent processed from higher dimensional automata" Lecture Notes in Computer Science. Springer-Verlag, 1059. 72-86 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] N.Ikeda,S.Manabe: "Van Vleck-Pauli formula for Wiener integrals and Jacobi fields" Ito's stochastic calculus and probability theory.Springer. 141-156 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] N.Ikeda,S.Kusuoga: "Short time asymptotics for fundamental solutions of heat equations with boundary conditions" Proceedings of a Taniguchi International Workshop,World Scientific. (予定). (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] T.Yamada: "Principle values of Brownian local times and their related topics" Ito's stochastic calculus and probability theory.Springer. 413-422 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] M.Arai,J.Uchiyama: "Growth order of eigenfunctions of Schrodinger operators with potentials admitting some integral conditons I・・・General theory・・・" Publ.R.I.M.S.Kyoto Univ.32. 581-616 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] G.A.Elliott,R.Nest T.Natsume: "The Atiya-Singer index theoremas passage to the classical limit on quantum mechanics" Comm.Math.Phys.182. 505-533 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] Y.Takayama: "Extraction of concurrent processes from higher dimensional automata" Lecture Notes in Computer Science,Springer. 1059. 72-86 (1996)

    • Related Report
      1996 Annual Research Report

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Published: 1996-04-01   Modified: 2016-04-21  

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