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Infinite dimensional and numerical stochastic analysis

Research Project

Project/Area Number 11440034
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 InstitutionKYUSHU UNIVERSITY

Principal Investigator

SUGITA Hiroshi  Faculty of Mathematics, Kyushu University, Ass. Prof., 大学院・数理研究院, 助教授 (50192125)

Co-Investigator(Kenkyū-buntansha) TAKANOBU Satoshi  Kanazawa Univ. Graduate Sch. Nat. Sci., Ass. Prof., 大学院・自然科学研究科, 助教授 (40197124)
OGAWA Shigeyoshi  Kanazawa Univ. Dept. of Eng., Prof., 工学部, 教授 (80101137)
TANIGUCHI Setsuo  Faculty of Mathematics, Kyushu University, Prof., 大学院・数理研究院, 教授 (70155208)
SATO Yoshihiro  GIFU-Syotoku Gakuen Univ., Dept. of Economic Information, Ass. Prof., 経済情報学部, 助教授 (30249213)
FUKAI Yasunari  Faculty of Mathematics, Kyushu University, Assistant, 大学院・数理研究院, 助手 (00311837)
Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥4,200,000 (Direct Cost: ¥4,200,000)
Fiscal Year 2001: ¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2000: ¥2,100,000 (Direct Cost: ¥2,100,000)
KeywordsInfinite dimensional stochastic analysis / Malliavin calculus / Stochastic numerics / Pseudorandom number generator / Numerical integration / Monte-Carlo integration / Random Weyl sampling / finite integral adeles / 無限次元確率解析 / 擬似乱数 / アデール環 / ランダムサンプリング / pairwise independent / 確率振動積分
Research Abstract

1. A fundamental inequality for random sampling methods : A fundamental inequality for random sampling methods, which shows the limitation of accuracy of each method, has been obtained. According to the inequality, if we admit very complicated integrands, for any random sampling method, there exists an integrand for which the sampling method is as good as the i.i.d.-sampling or worse than it. In this sense, the i.i.d.-sampling is a stable numerical integration method for any complicated integrands.
2. Development of the random Weyl sampling : The random Weyl sampling, a random sampling method which requires much less randomness than i.i.d.-sampling, but is applicable for any complicated integrands, has been developed. It is a method by means of pairwise independent random samples. This reduction of randomness has the following advantages : (1) It is insensitive to the quality of pseudo-random generators. So, we can be careless in choosing pseudo-random generators. (2) The speed of generating samples is very rapid and it is almost independent of the speed of the pseudo-random generator in use. So, we can use a precise but slow pseudo-random generator, and then, a very reliable numerical integration for complicated integrands is possible.
3. Improvement of number theoretic density theorems by means of adeles : Number theoretic density theorems are not probabilistic objects by themselves. But we discovered that by using the ring of finite integral adeles(, which is an infinite dimensional extension of the ring of integers,) and the natural uniform probability measure on it, those density theorems can be formulated as proper probabilistic theorems, that is, law of large numbers. Moreover, we succeeded in describing the limit distributions of central limit theorem-scaled functions for a certain example.

Report

(4 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (54 results)

All Other

All Publications (54 results)

  • [Publications] Sugita, Hiroshi: "Dynamic random Weyl sampling for drastic reduction of randomness in Monte-Carlo integration"Proc. of IMACS 2001. (発表予定). (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kubota, Hisayoshi, Sugita, Hiroshi: "Probabilistic proof of limit theorems in number theory by means of adeles"Kyushu J. Math.. (発表予定). (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Sugita, Hiroshi: "Robust numerical integration and pairwise independent random variables"Jour. Comput. Appl. Math.. 139. 1-8 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Sugita, Hiroshi, Takanobu, Satoshi: "A limit theorem for Weyl transformation in infinite-dimensional torus and central limit theorem for correlated multiple Wiener integrals"J. Math. Sci. Univ. Tokyo. 7・1. 99-146 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Sugita, Hiroshi, Takanobu, Satoshi: "Random Weyl sampling for robust numerical integration of complicated functions"Monte Carlo Methods Appl.. 6・1. 27-48 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Sugita, Hiroshi, Taniguchi, Setsuo: "a remark on stochastic oscillatory integrals with respect to a pinned Wiener measure"Kyushu J. Math.. 53・1. 151-162 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Taniguchi, Setsuo: "Exponential decay of stochastic oscillatory integrals on classical Wiener spaces"J. Math. Soc. Japan. (発表予定). (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Taniguchi, Setsuo et al.: "Ground state estimations in gauge theory"Bull. Sci. math.. 125. 623-640 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Taniguchi, Setsuo: "Levy's stochastic area and the principle of stationary phase"J. Funct. Anal.. 172・1. 165-176 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Taniguchi, Setsuo: "Stochastic oscillatory integrals with quadratic phase function and Jacobi equations"Probab. Theory Related Fields. 114・3. 291-308 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Ishiwata, Akira, Taniguchi, Setsuo: "On the analyticity of stochastic flows"Osaka J. Math.. 36・1. 139-148 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Oagawa, Shigeyoshi: "On a deterministic approach to the numerical solution of the SDE ---Application of pseudo random numbers and functions"Math. and Comput. in Simulation. (発表予定). (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 小川重義: "確率微分方程式の数値解法"数学,岩波書店. 53・1. 34-45 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 小川重義, 金川秀也: "確率微分方程式の数値解法,2-応用編"数学,岩波書店. 53・2. 125-138 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Ogawa, Shigeyoshi: "On a class of SPDEs called Brownian particle equation---model for nonlinear diffusions. Monte Carlo and probabilistic methods for partial differential equations, Part II"(Monte Carlo, 2000) Monte Carlo Methods Appl.. 7・4. 321-328 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Ichinose, Takashi, Takanobu, Satoshi: "The norm estimate of the difference between the Kac operator and Schrodinger semigroup. II. The general case including the relativistic case"Electron. J. Probab.. 5・4. 1-47 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Fukai, Yasunari: "Hitting distribution to a quadrant of two-dimensional random walk"Kodai Math. J.. 23・1. 35-80 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 小川重義, 森眞: "現象から学ぶ確率論入門"講談社サイエンティフィック. 166 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Sugita: "Dynamic random Weyl sampling for drastic reduction of randomness in Monte-Carlo integration"Proc. of IMACS 2001. (to appear). (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Kubota and H. Sugita: "Probabilistic proof of limit theorems in number theory by means of adeles"Kyushu J. Math.. (to appear). (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Sugita: "Robust numerical integration and pairwise independent random variables"Jour. Comput. Appl. Math.. 139. 1-8 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Sugita and S. Takanobu: "A limit theorem for Weyl transformation in infinite-dimensional torus and central limit theorem for correlated multiple Wiener integrals"J. Math. Sci. Univ. Tokyo. 7. 99-146 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Sugita and S. Takanobu: "Random Weyl sampling for robust numerical integration of complicated functions"Monte Carlo Methods Appl.. 6. 27-48 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Sugita and S. Taniguchi: "A remark on stochastic oscillatory integrals with respect to a pinned Wiener measure"Kyushu J. Math.. 53. 151-162 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Taniguchi: "Exponential decay of stochastic oscillatory integrals on classical Wiener spaces"J. Math. Soc. Japan.. (to appear). (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Taniguchi et al.: "Ground state estimations in gauge theory"Bull. Sci. math.. 125. 623-640 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Taniguchi: "Levy's stochastic area and the principle of stationary phase"J. Funct. Anal.. 172. 165-176 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Taniguchi: "Stochastic oscillatory integrals with quadratic phase function and Jacobi equations"Probab. Theory Related Fields. 114. 291-308 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] A. Ishiwata and S. Taniguchi: "On the analyticity of stochastic flows"Osaka J. Math.. 36. 139-148 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Ogawa: "On a deterministic approach to the numerical solution of the SDE - Application of pseudo random numbers and functions"Math. and Comput. in Simulation. (to appear). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Ogawa: "Numerical solution of SDE (Japanese)""Suugaku", Iwanami. 53. 34-45 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Kanagawa and S. Ogawa: "Numerical solution of SDE, 2-Applications (Japanese)""Suugaku", Iwanami. 53. 125-138 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Ogawa: "On a class of SPDEs called Brownian particle equation - model for nonlinear diffusions. Monte Carlo and probabilistic methods for partial differential equations, Part II"(Monte Carlo, 2000) Monte Carlo Methods Appl.. 7. 321-328 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Ichinose and S. Takanobu: "The norm estimate of the difference between the Kac operator and Schrodinger semigroup. II. The general case including the relativistic case"Electron. J. Probab.. 5. 1-47 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y. Fukai: "Hitting distribution to a quadrant of two-dimensional random walk"Kodai Math. J.. 23. 35-80 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Sugita, Hiroshi: "Dynamic random Weyl sampling for drastic reduction of randomness in Monte-Carlo integration"Proc.of IMACS 2001. (発表予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kubota, Hisayoshi, Hiroshi, Sugita: "Probabilistic proof of limit theorems in number theory by means of adeles"Kyushu J.Math. (発表予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Sugita, Hiroshi: "Robust numerical integration and pairwise independent random variables"Jour.Comput.Appl.Math.. 139. 1-8 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Taniguchi, Setsuo: "Exponential decay of stochastic oscillatory integrals on classical Wiener spaces"J.Math.Soc.Japan. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] Taniguchi, Setsuo et al.: "Ground state estimations in gauge theory"Bull.Sci.math.. 125. 623-640 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Taniguchi, Setsuo: "Analytic functions on abstract Wiener spaces"J.Funct.Anal.. 179. 235-250 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 小川重義, 森 眞: "現象から学ぶ確率論入門"講談社サイエンティフィック. 166 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Sugita,Hiroshi: "Robust numerical integration and pairwise independent random variables "Jour.Comput.Appl.Math.. (発表予定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Sugita,Hiroshi and Takanobu,Satoshi: "A limit theorem for Weyl transformation in infinite-dimensional torus and central limit theorem for correlated multiple Wiener integrals"J.Math.Sci.Univ.Tokyo. 7・1. 99-146 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Sugita,Hiroshi and Takanobu,Satoshi: "Random Weyl sampling for robust numerical integration of complicated functions"Monte Carlo Methods Appl.. 6・1. 27-48 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Ichinose,Takashi and Takanobu,Satoshi: "The norm estimate of the difference between the Kac operator and Schrodinger semigroup.II. The general case including the relativistic case"Electron.J.Probab.. 5・4. 1-47 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Taniguchi,Setsuo: "Levy's stochastic area and the principle of stationary phase"J.Funct.Anal.. 172・1. 165-176 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Fukai,Yasunari: "Hitting distribution to a quadrant of two-dimensional random walk"Kodai Math.J.. 23・1. 35-80 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Hiroshi Sugita and Satoshi Takanobu: "A limit the orem for weyl transformation in inginite-dimen-sional torus and C.L.T.for correlated multiple wiener integrals"Lournal of Mathematical Sciences,Univ.of Tokyo. 印刷中. (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Hiroshi Sugita and Setsuo Taniguchi: "A remark on stochastic oscillatory integrals with respect to a pinned Wiener measure"Kyushu Journal of Mathematics. 53・1. 151-162 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Setsuo Taniguchi: "Stochastic oscillatory integvels with quedratic phase function and jacobi cquation"Profbility Theory and Releted Fields. 114・3. 291-308 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Setsuo Taniguchi: "Levy's stochastic avea and the principle of stationary phase"Journal of Funvtianal Analysis. (印刷中).

    • Related Report
      1999 Annual Research Report
  • [Publications] Takashi Ichinose and Satoshi Takanobu: "The novm estimate of the difference between the Kac operctor and Schrodinger semigreupII:The general case"Electoronic Journal of Prefability. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] Yasunari Fukai: "Hitting distribation to a quadrant of two-dinebsuibal random walk"Kodai Mathematical Journal. (発表予定).

    • Related Report
      1999 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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