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Constructive research on infinite dimensional stochastic Processes and its applications to quantum information analysis

Research Project

Project/Area Number 19540201
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionMeijo University

Principal Investigator

SAITO Kimiaki  Meijo University, 理工学部, 教授 (90195983)

Co-Investigator(Kenkyū-buntansha) 松澤 忠人  名城大学, 理工学部, 教授 (20022618)
尾畑 伸明  東北大学, 大学院・情報科学研究科, 教授 (10169360)
日比野 雄嗣  佐賀大学, 理工学部, 准教授 (50253589)
三町 祐子  名城大学, 理工学部, 助教 (00218629)
西 健次郎  名城大学, 理工学部, 講師 (30076616)
Co-Investigator(Renkei-kenkyūsha) MATSUZAWA Tadato  名城大学, 理工学部, 教授 (20022618)
OBATA Nobuaki  東北大学, 大学院・情報科学研究科, 教授 (10169360)
HIBINO Yuji  佐賀大学, 理工学部, 准教授 (50253589)
MIMACHI Yuko  名城大学, 理工学部, 助教 (00218629)
NISHI Kenjiro  名城大学, 理工学部, 講師 (30076616)
Project Period (FY) 2007 – 2008
Project Status Completed (Fiscal Year 2008)
Budget Amount *help
¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Fiscal Year 2008: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2007: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords確率解析 / 無限次元解析 / 量子情報論 / 無限次元確率過程 / 量子確率過程 / 無限次元ラプラシアン / 量子ラプラシアン / フラクショナルブラウン運動 / 汎関数微分方程式 / ファイナンス
Research Abstract

平成19年度、20年度に上記補助金を受けさせて頂き、課題にある研究を遂行させて頂いた。本研究の目的は、確率論、解析学、グラフ理論、数論、計算機科学のそれぞれの専門分野から総合的に応用に適した無限次元確率解析を共同研究し、その立場からの量子化法、さらに量子情報解析として新しいアプローチを展開することにある。主に、無限次元ラプラシアン、特にレヴィ・ラプラシアンの一般化(エキゾチック・ラプラシアンとよばれる)に基づいた無限次元確率解析およびその量子確率過程の構成法への応用において成果があがった。この作用素エキゾチック・ラプラシアンとグロスラプラシアンとの相似性の証明からこの作用素の生成する無限次元ブラウン運動の構成を行ない、成果は国際雑誌IDAQPに掲載された。さらに、ホワイトノイズ超汎関数の空間は異なるオーダーをもつ無限個のエキゾチック・ラプラシアンを用いて分解できるという結果も得ることができた。エキゾチック・ラプラシアンはそれぞれ無限次元ブラウン運動に結合していることから、分解された空間一つ一つに異なる無限次元ブラウン運動が対応するという興味深い特徴づけができることになる。これらの成果と以前からの代表者の確率過程構成法との関連性の研究を進めている。また、抽象ウィナー空間上でこれらの結果を展開し始め、ルイジアナ州立大学の郭教授との共同研究を始めている。ローマ・トルヴェルガタ大学ヴォルテラ研究所のアカルディ教授とは、この成果を基に共同研究として、新しいホワイトノイズ理論構築を進めている。このように、ホワイトノイズ理論がスタイルを変え始め、付随した量子ホワイトノイズ理論も展開できることになり、この路線に沿った量子情報解析の構築が可能になったことが本課題の成果の意義のある点である。

Report

(3 results)
  • 2008 Annual Research Report   Final Research Report ( PDF )
  • 2007 Annual Research Report
  • Research Products

    (32 results)

All 2009 2008 2007

All Journal Article (15 results) (of which Peer Reviewed: 14 results) Presentation (17 results)

  • [Journal Article] A Gauss-Poisson correspondence and the Levy Laplacian, to appear in Interdisciplinary2009

    • Author(s)
      K. Saito
    • Journal Title

      Information Sciences

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Analysis Of Markov chain with rewards by Z-transformation and the decision of itsoptimal policy2009

    • Author(s)
      Y. Uchimura and Y.H.-Mimachi
    • Journal Title

      International J. of Pureand Applied Math Vol.51, No.2

      Pages: 277-291

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Exotic Laplacians and associated stochastic processes2009

    • Author(s)
      L. Accardi, U. C. Ji, K. Saito
    • Journal Title

      Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 12(To appear)

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Exotic Laplacians and associated stochastic processes, Infinite Dimensional Analysis2008

    • Author(s)
      L. Accardi, U.C.Ji, K. Saito
    • Journal Title

      Quantum Probability and Related Topics Vol.10, No.2

      Pages: 1-19

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] A Gauss-Poisson correspondence and infinite dimensional Laplacians2008

    • Author(s)
      K. Saito
    • Journal Title

      Far East Journal of Mathematical Sciences Vol.31

      Pages: 31-47

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] An infinite dimensional Fourier-Mehler transform andthe Levy Laplacian2008

    • Author(s)
      K. Saito, K. Sakabe
    • Journal Title

      QP-PQ : Quantum Probability and White Noise Analysis Vol.22

      Pages: 149-160

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] A Gauss-Poisson Correspondence and infinite dimensional Laplacians2008

    • Author(s)
      K. Saito
    • Journal Title

      Far East Journal of Mathematical Sciences Vol. 31

      Pages: 31-47

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Infinite dimensional stochastic processes associated with the Exotic Laplacians2008

    • Author(s)
      K. Saito
    • Journal Title

      RIMS Kokyuroku'Non-Commutative Analysis and Micro-Macro Duality' No. 1609

      Pages: 49-62

    • Related Report
      2008 Annual Research Report
  • [Journal Article] An infinite dimensional Fourier-Mehler transform and the Levy Laplacian2008

    • Author(s)
      K. Saito, K. Sakabe
    • Journal Title

      QP-PQ: Quantum Probability and White Noise Analysis Vol.21

      Pages: 149-160

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Levy Laplacianacting on some class of Levy functionals2007

    • Author(s)
      K. Saito, A.H. Tsoi
    • Journal Title

      QP-PQ : Quantum Probability and White Noise Analysis Vol.20

      Pages: 330-337

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] A similarity between the Gross Laplacian and the Levy Laplacian, Infinite Dimensional Analysis2007

    • Author(s)
      U.C. Ji, K. Saito
    • Journal Title

      Quantum Probability and Related Topics Vol.10, No.2

      Pages: 261-276

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Quantum stochastic process associated with Quantum LevyL aplacian2007

    • Author(s)
      U.C. Ji, K. Saito, H. Ouerdiane
    • Journal Title

      Communications on Stochastic Analysis Vol.1, No.2

      Pages: 231-245

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] The Levy Laplacian acting on some class of Levy functionals2007

    • Author(s)
      K. Saito, A. H. Tsoi:
    • Journal Title

      QP-PQ: Quantum Probability and White Noise Analysis Vol.20

      Pages: 330-337

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A similarity between the Gross Laplacian and the Levy Laplacian2007

    • Author(s)
      U. C. Ji, K. Saito
    • Journal Title

      Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol.10, No.2

      Pages: 261-276

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Quantum stochastic process associated with Quantum Levy Laplacian2007

    • Author(s)
      U. C. Ji, K. Saito, H. Ouerdiane
    • Journal Title

      Communications on Stochastic Analysis Vol.1, No.2

      Pages: 231-245

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Presentation] A Gauss-Poisson correspondence and infinite dimensional Laplacians2008

    • Author(s)
      K. Saito
    • Organizer
      1037th AMS meeting
    • Place of Presentation
      Louisiana State Univ., USA
    • Year and Date
      2008-03-29
    • Related Report
      2007 Annual Research Report
  • [Presentation] Exotic Laplacians and exponential functions of white noise2008

    • Author(s)
      K. Saito
    • Organizer
      The 8th Sendai Workshop on Infinite Dimensional Analysis and Quantum Probability
    • Place of Presentation
      Tohoku University
    • Related Report
      2008 Final Research Report
  • [Presentation] Exotic Laplacians and associated stochastic processes2008

    • Author(s)
      K. Saito
    • Organizer
      Invited Lecture
    • Place of Presentation
      Chungbuk National University, Korea
    • Related Report
      2008 Final Research Report
  • [Presentation] Infinite dimensionalstochastic processes associated with theExotic Laplacians2008

    • Author(s)
      U.C.Ji, K. Saito
    • Organizer
      数理解析研究所講究録
    • Place of Presentation
      Non-Commutative Analysis and Micro-Macro Duality
    • Related Report
      2008 Final Research Report
  • [Presentation] Exotic Laplacians and associated stochastic processes2008

    • Author(s)
      K. Saito
    • Organizer
      Invited Lecture, Univ. Rome
    • Place of Presentation
      Volterra Center, Italy
    • Related Report
      2008 Final Research Report
  • [Presentation] A Gauss-Poisson correspondence and infinite dimensional Laplacians, AMS meetings2008

    • Author(s)
      K. Saito
    • Organizer
      Invited Talk
    • Place of Presentation
      Louisiana State University, USA, Abstract
    • Related Report
      2008 Final Research Report
  • [Presentation] Exotic Laplacians and exponential functions of white noise2008

    • Author(s)
      K. Saito
    • Organizer
      The 8^<th> Sendai Workshop on Infinite Dimensional Analysis and Quantum Probability
    • Place of Presentation
      東北大学大学院情報科学研究科
    • Related Report
      2008 Annual Research Report
  • [Presentation] Exotic Laplacians and associated stochastic processes2008

    • Author(s)
      K. Saito
    • Organizer
      Invited Lectures on Infinite dimensional Analysis Chungbuk National University, Korea
    • Place of Presentation
      Chungbuk National University
    • Related Report
      2008 Annual Research Report
  • [Presentation] Stochastic processes generated by the Exotic Laplacians, RIMS研究集会2007

    • Author(s)
      K. Saito
    • Organizer
      非可換解析とミクロ・マクロ双対性
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2008 Final Research Report
  • [Presentation] An infinite dimensional Fourier-Mehler transform and the Levy Laplacian2007

    • Author(s)
      K. Saito
    • Organizer
      Invited Lecture at National Chungbuk University
    • Place of Presentation
      Korea
    • Related Report
      2008 Final Research Report
  • [Presentation] Infinite dimensional Schrodinger equation associated with the Levy and Volterra Laplacians2007

    • Author(s)
      K. Saito
    • Organizer
      Invited Lecture at Dankook University
    • Place of Presentation
      Korea
    • Related Report
      2008 Final Research Report
  • [Presentation] Stochastic processes associated with the Exotic Laplacians2007

    • Author(s)
      K. Saito
    • Organizer
      Invited Lecture at Hanyan University
    • Place of Presentation
      Korea
    • Related Report
      2008 Final Research Report
  • [Presentation] Exotic Laplacians and associated stochastic processes2007

    • Author(s)
      K. Saito
    • Organizer
      28^<th> Conference on Quantum Probability and Related Topics
    • Place of Presentation
      CIMAT, Mexico
    • Related Report
      2008 Final Research Report
  • [Presentation] Recent developments on theLevy Laplacian and related stochastic processes2007

    • Author(s)
      K. Saito
    • Organizer
      The 6^<th> Sendai Workshopon Infinite Dimensional Analysis and Quantum Probability under the auspices of JSPS- KOSEF Joint Program
    • Place of Presentation
      Tohoku University
    • Related Report
      2008 Final Research Report
  • [Presentation] Kuo's Fourier-Mehler transform and the Levy Laplacian, AMS-MAA Joint Mathematics Meetings2007

    • Author(s)
      K. Saito
    • Organizer
      New Orleans
    • Place of Presentation
      Abstracts of papers presented to the American Mathematical Society
    • Related Report
      2008 Final Research Report
  • [Presentation] Infinite dimensional stochastic processes generated by the Exotic Laplacians2007

    • Author(s)
      K. Saito
    • Organizer
      RIMS研究集会「非可換解析とミクロ・マクロ双対性」
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2007 Annual Research Report
  • [Presentation] Exotic Laplacians and associated stochastic processes2007

    • Author(s)
      K. Saito
    • Organizer
      28th Conference on Quantum Probability and Related Topics
    • Place of Presentation
      CIMAT, Mexico
    • Related Report
      2007 Annual Research Report

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Published: 2007-04-01   Modified: 2016-04-21  

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