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Infinite Dimensional Stochastic Analysis and its Applications

Research Project

Project/Area Number 11640139
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 InstitutionMeijo University

Principal Investigator

SAITO Kimiaki  Meijo Univ., Dep.Info.Sci., Prof., 理工学部, 教授 (90195983)

Co-Investigator(Kenkyū-buntansha) NISHI Kenjiro  Meijo Univ., Dep.Info.Sci., Assi.Prof., 理工学部, 講師 (30076616)
SI Si  Aichi Pref.Univ., Dep.Info.Sci., Asso.Prof., 情報科学部, 助教授 (70269687)
HIDA Takeyuki  Meijo Univ., Institute., Visiting Prof., 総合研究所, 客員教授 (90022508)
MIMACHI Yuko  Meijo Univ., Dep.Info.Sci., Assi.Prof., 理工学部, 講師 (00218629)
Project Period (FY) 1999 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 2000: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1999: ¥1,100,000 (Direct Cost: ¥1,100,000)
KeywordsInfinite Dimensional Stochastic Analysis / White Noise Theory / Levy Laplacian / Infinite Dimensional Stochastic Process / Quantum Stochastic Process / Quantum Information / レヴィ・テプラシアン / ホワイトノイズ / Levy's Laplacian / 加法過程
Research Abstract

Supported by Grant-in-Aid for Scientific Research(C)we obtained fruitful results on an infinite dimensional stochastic analysis based on the Levy Laplacian. The Laplace equation associated the Levy Laplacian is equivalent to the yang-Mills equation in quantum physics. Moreover, the Laplacian is closely related to the square power of the quantum white noise. To research a stochastic analysis based on the Laplacian has an important role of describing and understanding the micro-phenomena. With our previous researches in 1999 we obtained several results on a stochastic analysis based on the Laplacian in this research : Diagnalization of the Levy Laplacian, Constructing the domain(nuclear space)on which the Laplacian has a continuous spectrum, Constructing a stochastic process and a quantum stochastic process generated by the Laplacian on the domain, Constructing a stochastic process generated by functions of the Laplacian, etc. Those stochastic processes are infinite but the parts of fluctuation are finite and therefore we expect to apply this theory to other fields in Engineering. Moreover we can describe some biological phenomena and economical phenomena as stochastic models using the Levy Laplacian. If we change elements of the domain of the Levy Laplacian to some operators, then we can research the quantum white noise theory based on the Laplacian. We hope that this direction of researches has more furuitful developments.
Supported by the Grant we attended at international workshops in Italy and in USA to give lectures on recent results on infinite dimensional stochastic analysis, and also gave talks at Kyoto University, Kyushu University and others. We asked Prof.Ito at Hiroshima Women's University to give a review on applications of white noise theory to the economy.

Report

(3 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • Research Products

    (29 results)

All Other

All Publications (29 results)

  • [Publications] K.Saito, A.H.Taoi: "Stochastic processes Generated by functions of the Levy Laplacian"Quantum Information II. Vol.2. 183-194 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Saito: "The Levy Laplacian and Stochastic Processes"Infinite Dimensional Harmonic Analysis. Vol.2. 306-318 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Saito: "A Stochastic Process generated by the Levy Laplacian"Acta Applicandae Mathematicae. (掲載予定). (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] D.M.Chung, U.C.Ji, K.Saito: "Notes on a Co-group generated by the Levy Laplacian"Proc.American Mathematical Society. (掲載予定). (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Nishi, K.Saito, A.H.Tsoi: "A Stochastic Expression of a Semi-Group generated by the Levy Laplacian"Quantum Information III. (掲載予定). (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Saito: "The Levy Laplacian and Stable Processes."Chaos, Solitons & Fractals/Les Treilles Special Issue. (掲載予定). (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Saito: "A stochastic process generated by the Levy Laplacian"Volterra International School Trento, Italy Acta Appl.Math.. (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Saito: "The Levy Laplacian and Stochastic Processes"Infinite Dimensional Harmonic Analysis. Vol.2(D.+M.Grabner). 306-318 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Saito, A.H.Tsoi: "The Levy Laplacian as a self-adjoint operator"Quantum Information World Scientific. 159-171 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Saito, A.H.Tsoi: "The Levy Laplacian acting on Poisson noise functionals"Infinite Dimensional Analysis, Quantum Probability and Related Topics. Vol.2. 503-510 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Saito: "Infinite dimensional stochastic processes generated by extensions of the Levy Laplacian"Publication of Centro Vito Volterra. N.428. 1-12 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Saito, A.H.Tsoi: "Stochastic processes generated by functions of the Levy Laplacian"Quantum Information II World Scientific. 183-194 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Saito: "The Levy Laplacian and stable processes"Chaos, Solitons & Fractals/Les Treilles Special Issue, International Solvay Institute. (to appear). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] D.M.Chung, U.C.Ji, K.Saito: "Notes on a C_0-group generated by the Levy Laplacian"Proc.American Mathematical Society. (to appear). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Nishi, K.Saito, A.H.Tsoi: "A stochastic expression of a semi-group generated by the Levy Laplacian"Quantum Information III World Scientific. (to appear). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Saito: "Semigroups and Stochastic Processes associated with functions of the Levy Laplacian"RIMS Kokyuroku. 1157. 101-114 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Nkshi, K.Saito: "The Levy Laplacian and the Number operator"RIMS Kokyuroku. 1193(to appear). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Saito,A.H.Tsoi: "Stochastic processes generated by functions of the Levy Laplacian"Quantum Information II. Vol.2. 183-194 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Saito: "The Levy Laplacian and Stochastic Processes"Infinite Dimensional Harmonic Analysis. Vol.2. 306-318 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Saito: "A Stochastic Process generated by the Levy Laplacian"Acta Applicandae Mathematicae. (掲載予定). (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] D.M.Chung,U.C.Ji,K.Saito: "Notes on a Co-group generated by the Levy Laplacian"Proc.American Mathematical Society. (掲載予定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Nishi,K.Saito,A.H.Tsoi: "A Stochastic Expression of a Semi-Group generated by the Levy Laplacian"Quantum Information III. (掲載予定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Saito: "The Levy Laplacian and Stable Processes."Chaos,Solitons & Fractals / Les Treilles Special Issue. (掲載予定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] K. Saito: "The Levy Laplacian as a self-adjoint operator"Quantum Information, world Scientific Pub.. Vol.1. 159-171 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K. Saito: "The Levy Laplacian acting on Poisson Noise Functionals"Infinite Dim. Anal., Quantum Prob. and Related Topics. Vol.2,No.4. 503-510 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K. Saito: "The Levy Laplacian and stable processes"Chaos, Solitons & Fractals/Les Treilles Special Issue. (掲載予定). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] K. Saito: "A Stochastic process generated by the Levy Laplacian"Recent Developments in Infinite Dim. Anal. and Q.P.. (掲載予定). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] K. Saito: "Stochastic Processes generated by functions of the Levy Laplacion"Quantum Information II, world Scientific Pub.. (掲載予定). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] K. Saito: "The Levy Laplacian and Stochastic Processes"Infinite Dimensional Harmonic Analysis. (掲載予定). (2000)

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2021-04-07  

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