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New developments on quantum information theory by an infinite dimensional stochastic analysis and applications to dynamical system theory

Research Project

Project/Area Number 24540149
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionMeijo University

Principal Investigator

SAITO KIMIAKI  名城大学, 理工学部, 教授 (90195983)

Co-Investigator(Kenkyū-buntansha) MIMACHI Yuko  名城大学, 理工学部, 准教授 (00218629)
Co-Investigator(Renkei-kenkyūsha) ICHIHARA Kanji  名城大学, 理工学部, 教授 (00112293)
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, 工学系研究科, 准教授 (50253589)
Project Period (FY) 2012-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords無限次元解析 / 確率解析 / 関数解析 / 量子情報論 / 力学系理論 / 無限次元ラプラシアン / 確率論的数論 / 量子確率論 / 無限次元確率過程 / 数論的確率論 / 分枝過程 / 量子ホワイトノイズ / 無限次元ラプラス作用素 / 無限次元確率解析 / 量子情報解析 / 量子確率過程 / 超函数の積 / 無限次元ブラウン運動の積 / 超汎関数空間 / 符号力学系 / エルゴード理論
Outline of Final Research Achievements

①We introduced an infinite dimensional Brownian motions associated with higher real order derivatives of white noise and proved that their generators of this stochastic processes are given by exotic Laplacians. We also proved that higher real order Cesaro Theorem, and that the exotic Laplacian coincides with the Levy Laplacian associated with higher real order derivative of white noise. ② Constructing a space of distributions on which the product is well-defined, we formulated powers of an infinite dimensional Brownian motion without any renormalization as mathematics. This implies a generalization of the Ito formula in stochastic analysis. On the space of distributions the divergent part in the theory of quantum field is also formulated as mathematics. ③ We gave a method of triangulation decomposition of the transition probability matrix of the G-W branching process, and obtained a limiting probability distribution of the process. We also developed a new algorism of the method.

Report

(4 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (13 results)

All 2015 2014 2013 2012

All Journal Article (7 results) (of which Peer Reviewed: 6 results,  Open Access: 1 results,  Acknowledgement Compliant: 1 results) Presentation (6 results) (of which Invited: 6 results)

  • [Journal Article] Asymptotic behavior of the Bernoulli type Galton–Watson branching process with immigration2015

    • Author(s)
      Kimiaki Saito, Yoshinori Uchimura
    • Journal Title

      Random Operators and Stochastic Equations

      Volume: 23 Pages: 1-10

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Powers of an infinite dimensional Brownian motion associated with the product of distributions2014

    • Author(s)
      Kimiaki Saito, Takashi Shimada
    • Journal Title

      Communications on Stochastic Analysis

      Volume: 8 Pages: 343-364

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Infinite dimensional Laplacians associated with derivatives of white noise2013

    • Author(s)
      Kimiaki Saito;
    • Journal Title

      Quantum Probability and White Noise Analysis

      Volume: Vol. 29 Pages: 233-248

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] The exotic Laplacians generate the Markov processes given by distribution derivatives of white noise2013

    • Author(s)
      Luigi Accardi, Un Cig Ji, Kimiaki Saito
    • Journal Title

      Infinite Dimensional Analysis, Quantum Probability and Related Topics

      Volume: Vol. 16 Issue: 03 Pages: 1-26

    • DOI

      10.1142/s0219025713500203

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Infinite dimensional Laplacians associated with derivatives of white noise2013

    • Author(s)
      Kimiaki Saito;
    • Journal Title

      Quantum Probability and Related Topics

      Volume: 29 Pages: 233-248

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] The exotic (Higher order Levy) Laplacians generate the Markov processes given by distribution derivatives of white noise2013

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

      Publication, Centro Vito Volterra

      Volume: 637 Pages: 1-25

    • Related Report
      2012 Research-status Report
  • [Journal Article] Limiting distributions of Galton-Watson branching processes with immigration2012

    • Author(s)
      Yoshinori Uchimura, Kimiaki Saito;
    • Journal Title

      Communications on Stochastic Analysis

      Volume: 6 Pages: 282-295

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Presentation] The product of distributions and renormalizations2015

    • Author(s)
      Kimiaki Saito, Takashi Shimada
    • Organizer
      Probability Seminar, Chungbuk National University
    • Place of Presentation
      Research Institute of Mathematical Finance, Chungbuk National University, Korea
    • Year and Date
      2015-02-26 – 2015-03-02
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] The product of distributions and powers of an infinite dimensional Brownian motion2014

    • Author(s)
      Kimiaki Saito
    • Organizer
      Joint research on Infinite Dimensional Analysis and Quantum Probability, Centro Vito Volterra
    • Place of Presentation
      Centro Vito Volerra, University of Rome Torvergata, Rome
    • Year and Date
      2014-05-01 – 2014-05-07
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Infinite dimensional analysis based on higher order derivatives of white noise2013

    • Author(s)
      Kimiaki Saito
    • Organizer
      The 8th Jikji Workshop Infinite Dimensional Analysis and Quantum Probability
    • Place of Presentation
      NIMS, Daejeon, Korea
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Infinite dimensional analysis based on higher order derivatives of white noise2012

    • Author(s)
      Kimiaki Saito
    • Organizer
      4th International Workshop on Quantum Probability and its Applications
    • Place of Presentation
      Campobasso; Italy,
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Infinite dimensional Laplacians associated with derivatives of an infinite dimensional Brownian motion2012

    • Author(s)
      Kimiaki Saito
    • Organizer
      Seminar on IDAQP, Centro Vito Volterra
    • Place of Presentation
      Rome, Italy
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Infinite dimensional Brownian motion associated with exotic Laplacian2012

    • Author(s)
      Kimiaki Saito
    • Organizer
      Seminar on Probability, Chungbuk National University
    • Place of Presentation
      Chungbuk National University
    • Related Report
      2012 Research-status Report
    • Invited

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Published: 2013-05-31   Modified: 2019-07-29  

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