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A study of identification problem for continuous model in phenomena of complex system based on the theory of Langevin equations from the viewpoint of the theory of stochastic processes

Research Project

Project/Area Number 10440026
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 InstitutionUniversity of Tokyo

Principal Investigator

OKABE Yasunori  Graduate School of Engineering, University of Tokyo, Professor, 大学院・工学系研究科, 教授 (30028211)

Co-Investigator(Kenkyū-buntansha) INOUE Akihiko  Graduate School of Science, Hokkaido University, Associate Professor, 大学院・理学研究科, 助教授 (50168431)
AIHARA Kazuyuki  Graduate School of Frontier Sciences, University of Tokyo, Professor, 大学院・新領域創成科学研究科, 教授 (40167218)
YANAGAWA Takashi  Graduate School of Science, Kyushu University, Professor, 大学院・数理学研究院, 教授 (80029488)
MATSUURA Masaya  Graduate School of Engineering, Assistant Professor, 大学院・工学系研究科, 助手
HORITA Takehiko  Graduate School of Engineerin, Lecturer, 大学院・工学系研究科, 講師 (90222281)
伏見 正則  東京大学, 大学院・工学系研究科, 教授 (70008639)
Project Period (FY) 1998 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥6,700,000 (Direct Cost: ¥6,700,000)
Fiscal Year 2000: ¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 1999: ¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 1998: ¥2,400,000 (Direct Cost: ¥2,400,000)
KeywordsKM_2O-Langevin equation / KMO-Langevin equation / weak stationarity / purely non-determinicity / fluctuation-dissipation theorem / local and global canonical representation theorem / innovation method / outer matrix function / 純非決定性,完全ランク性とテープリッツ条件 / 連続系と離散系 / 多次元非線形予測解析 / KMO-ランジュヴァンデータ / KM_2O-ランジュヴァンデータ / スケール極限 / 純非決定性完全ランク性とテープリッツ条件
Research Abstract

As a preparation for the aim to investigate weakly stationary process with continuous time by using weakly stationary process with discrete time, we developed the theory of KM_2O-Langevin equations for degenerate flows in the three directioins of the analysis of local non-linear information space, the analysis of weight transformations and the linear prediction theory. By using these results, we resolved not only the non-linear prediction problem for one-dimensional strictly stationary processes which had remained to be solved for a long time after Masani-Wiener's work, but also both the non-linear prediction problem and the non-linear filtering problem for multi-dimensional stochastic processes with discrete time.
On the other hand, we constructed Kubo noise associated with multi-dimensional stationary flow with continuous time in a Hilbert space.
Next, by taking a procedure of scaling limits of KM_2O-Langevin data that determines the local stochastic difference equation (KM_2O-Langevin … More equation) describing the time evolution of weakly stationary process with discrete time, we derived KMO-Langevin data that determines the global stochastic difference equation (KMO-Langevin equation). Conversely, for a class of weakly stationary process with continuous time, we derived KM_2O-Langevin equation from KMO-Langevin equation, by using the idea of innovation method in the filtering theory. Thus, we could obtain the algorithm calculating the discrete and global characteristics from the discrete and local characteristics and prove the representaion theorem of outer matrix function for continuous case from the one for discrete case. Therefore, we have completed the derivation of the stochastic difference equation describing the time evolution of weakly stationary process with discrete time not only for local case, but also for global case.
Moreover, for a class of weakly stationary process X=(X(t) ; t∈R), we define for each positive number ∈, we define a stochastic process X_∈=(X(n∈) ; n∈Z). Then, we investigated KMO(resp.KM_2O)-Langevin data that determines the dissipation term and the fluctuation term in KMO(resp.KM_2O)-Langevin equation describing the time evolution of the weakly stationary process X and certatin ∈-dependence of KMO(resp.KM_2O)-Langevin data that determines the dissipation term and the fluctuation term in KMO (resp.KM_<>O)-Langevin equation describing the time evolution of the weakly stationary process X_∈. In particular, we could represent KMO-Langevin data associated with X as a scaling limits with respect to ∈ of KMO-Langevin data associated with X_∈. Less

Report

(4 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • Research Products

    (27 results)

All Other

All Publications (27 results)

  • [Publications] Y.Okabe: "On a Kubo noise associated with a multi-dimensional stationary curve in a Hilbert space"Proceedings of the 2nd Jagna International Workshop, Mathematical Methods of Quantum Physics. 19-26 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Okabe and M.Matuura: "On the theory of KM_2O-Langevin equations for stationary flows (III) : extension theorem"Hokkaido Mathematical Journal. 29巻. 65-78 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Okabe and A.Kaneko: "On a non-linear prediction analysis for multi-dimensional stochastic processes with its applications to data analysis"Hokkaido Mathematical Journal. 29巻. 601-657 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] N.Masuda and Y.Okabe: "Time series analysis with wavelet coefficients"Japan Journal of Industrial and Applied Mathematics. 18巻. 131-160 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Matsuura and Y.Okabe: "On a non-linear prediction problem for one-dimensional stochastic processes"to appear in Japanese Jornal of Mathemasics. (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Okabe: "On the theory of KM_2O-Langevin equations for stationary flows (II) : construction theorem"to appear in the special volume in honor of the 70th birthday of Professor Takeyuki Hida. (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Okabe: "On a Kubo noise associated with a multidimensional stationary curve in a Hilbert space"Proceedings of the 2nd Jagna International Workshop, Mathematical Methods of Quantum Physics. (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Okabe and M.Matuura: "On the theory of KM_2O-Langevin equations for stationary flows (III) : extension theorem"Hokkaido Mathematical Journal. Vol.29. (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Okabe and A.Kaneko: "On a non-linear prediction analysis for multi-dimensional stochastic processes with its applications to data analysis"Hokkaido Mathematical Journal. Vol.29. (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] N.Masuda and Y.Okabe: "Time series analysis with wavelet coefficients"Japan Journal of Industrial and Applied Mathematics. (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Masuura and Y.Okabe: "On a non-linear prediction problem for one-dimensional stochastic processes"Japanese Journal of Mathematics. (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] Y.Okabe: "On the theory of KM_2O-Langevin equations for stationary flows (II) : construction theorem"the special volume in honor of the 70th birthday of Professor Takeyuki Hida. (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] Y.Okabe: "On a Kubo noise associated with a multidimensional stationary curve in a Hilbert space"Proceedings of the 2nd Jagna International Workshop,Mathematical Methods of Quantum Physics. 19-26 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] Y.Okabe and M.Matuura: "On the theory of KM_2O-Langevin equations for stationary flows(III) : extension theorem"Hokkaido Mathematical Journal. 29巻. 65-78 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Y.Okabe and A.Kaneko: "On a non-linear prediction analysis for multi-dimensional stochastic processes with its applications to data analysis"Hokkaido Mathematical Journal. 29巻. 601-657 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] N.Masuda and Y.Okabe: "Time series analysis with wavelet coefficients"to appear in Journal Japan Industrial and Applied Mathematical. (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Matsuura and Y.Okabe: "On a non-linear prediction problem for one-dimensional stochastic processes"to appear in Japanese Jornal of Mathemasics. (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Y.Okabe: "On the theory of KM_2O-Langevin equations for stationary flows(II) : construction theorem"to appear in the special volume in honor of the 70th birthday of Professor Takeyuki Hida. (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Y.Okabe and T.Yamane: "The theory of KM_2O-Langevin equation and its applications data analysis (III) : Deterministic analysis"Nagoya Mathematical Journal. 152巻. 175-201 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] Kanamaru,T.Horita and.Y.Okabe: "Stochastic resonance in the Hodgkin-Huxley network"Journal of the Physical Society of Japan. 67巻12号. 4058-4063 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] Y.Okabe: "On the theory of KM_2O-Langevin equation for stationary flows (I) : characterization theorem"Journal of the Physical Society of Japan. 51巻12号. 817-841 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 関本勝也,岡部靖憲,緒方純俊: "隣接する地域間における大気汚染物質の相互因果解析"計測自動制御学会論文集. 35巻12号. 1524-1529 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Y.Okabe: "On the theory of KM_2O-Langevin equation for stationary flows (II) : construction theorem"The special volume in honor of the 70th birthday of Professor Takeyuki Hida. (掲載予定). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Y.Okabe and M.Matsuura: "On the theory of KM_2O-Langevin equation for stationary flows (III) : extension theorem"Hokkaido Mathematical Journal. (掲載予定). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Y.Okabe and T.Yamane: "The theory of KM_2O-Langevin equations and its applications to data analysis(III):Deterministic analysis" Nagoya Mathematical Journal. 152号. 175-201 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Kanamaru,T.Horita and Y.Okabe: "Stochastic resonance in the Hodgkin-Huxley network" Journal of the Physical Society of Japan. 67巻12号. 4058-4063 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Okabe: "On the theory of KM_2O-Langevin equations for stationary flows(I):characterization theorem" Journal of the Mathematical Society of Japan. 52巻(掲載予定). (1999)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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