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A Study of asymptotic behaviors of stochastic oscillatory integrals

Research Project

Project/Area Number 11440051
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKYUSHU UNIVERSITY

Principal Investigator

TANIGUCHI Setsuo  Faculty of Mathematics, Kyushu University, Prof., 大学院・数理学研究院, 教授 (70155208)

Co-Investigator(Kenkyū-buntansha) YASUDA Kumi  Faculty of Mathematics, Kyushu University, Res. Ass., 大学院・数理学研究院, 助手 (40284484)
HAMANA Yuji  Faculty of Mathematics, Kyushu University, Ass. Prof., 大学院・数理学研究院, 助教授 (00243923)
SUGITA Hiroshi  Faculty of Mathematics, Kyushu University, Ass. Prof., 大学院・数理学研究院, 助教授 (50192125)
MATSUMOTO Hiroyuki  Nagoya Univ., Faculty of Information and culture, Ass. Prof., 情報分科学部, 助教授 (00190538)
FUKAI Yasunari  Faculty of Mathematics, Kyushu University, Res. Ass., 大学院・数理学研究院, 助手 (00311837)
國田 寛  九州大学, 大学院・数理学研究科, 教授 (30022552)
吉川 敦  九州大学, 大学院・数理学研究科, 教授 (80001866)
Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥8,800,000 (Direct Cost: ¥8,800,000)
Fiscal Year 2001: ¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2000: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 1999: ¥3,100,000 (Direct Cost: ¥3,100,000)
KeywordsStochastic oscillatory integral / Malliavin calculus / quadratic phase function / localization / asymptotic theory / heat kernel / trace formula / large deviation / 確率振動積分 / 確率微分方程式 / レヴィ測度 / ブラウン運動 / グリーン関数 / 非線形変換 / 複素変数変換 / ヤコビ方程式 / 確率面積 / 停留位相法
Research Abstract

In this research, we have made a systematic study on the asymptotic behavior of stochastic oscillatoty integrals. A stochastic oscillatory integral I(a) is, by definition, a integral of exp[iaq(x)]f(x) over the Wiener space X with respect to the Wiener measure on it, where i is the square root of -1, a is a real number, q, f are Wiener functionals on X. Obviously I(a) gives a characteristic function of the distribution of q under f(x)m(dx), and hence it is a basic object in the probability theory. Recalling the theory of Feynman path integrals, one recognizes the real interest of stochastic oscillatory integrals. Namely, a stochastic oscillatory integral is a mathematical counterpart to Feynman path integral, and the study of its asymptotic behavior closely relates to, so called, the WKB approximation, the semi-classical approximation, and so on. In our study, following the well developed theory of statinary phase method on finite dimensional spaces, we made several basic but indispensable researches on the asymptotic behavior of stochastic oscillatory integrals. We established several explicit representation of stochastic oscillatory integrals with quadratic phase functions, and apply them to show a principle of stationary phase for such oscillatory integrals. Moreover, we spelled out the relationship between the decay order of integrals and the quadratic phase functions. We also showed that a localization to stationary points of the main part of the asymptototic behavior occurs for some stochastic oscillatory integrals. We moreover made several concrete observations when the oscillatory integral is defined on the classical Wiener space, the path space.

Report

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

    (28 results)

All Other

All Publications (28 results)

  • [Publications] H.Sugita, 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(和文)」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(和文)」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(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Matsumoto: "Closed form formulae for the heat kernels and the Green functions for the Laplacians on the symmetric spaces of rank one"Bull. Sci. math.. 125. 553-581 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K.Yasuda: "Extension of measures to infinite dimensional spaces over p-adic field"Osaka J. Math.. 37. 967-985 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Kesten, Y.Hamana: "A large-deviation result for the range of random walk and for the Wiener sausage"Probability Theory and Related Fields. 120. 183-208 (2001)

    • Description
      「研究成果報告書概要(和文)」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: "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] 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] H. Matsumoto: "Closed form formulae for the heat kernels and the Green functions for the Laplacians on the symmetric spaces of rank one"Bull. Sci. math.. 125. 553-581 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K. Yasuda: "Extension of measures to infinite dimensional spaces over p-adic field"Osaka J. Math.. 37. 967-985 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Kesten and Y. Hamana: "A large-deviation result for the range of random walk and for the Wiener sausage"Probab. Theory Related Fields. 120. 183-208 (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: "Analytic functions on abstract Wiener spaces"J.Funct.Anal.. 179. 235-250 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] A.-B.Cruzeiro, P.Malliavin, S.Taniguchi: "Ground state estimations in gauge theory"Bull.Sci.math.. 125. 623-640 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Taniguchi: "Exponential decay of stochastic oscillatory integrals on classical Wiener spaces"J.Math.Soc.Japan. (掲載予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Matsumoto: "Closed form formulae for the heat kernels and the Green functions for the Laplacians on the symmetric spaces of rank one"Bull.Sci.math.. 125. 553-581 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Albeverio, W.Karwowski, K.Yasuda: "Trace formula for p-adics"Acta.Appl.Math.. (掲載予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Kesten, Y.Haman: "A large-deviation result for the range of random walk and for the Wiener sausage"Probability Theory and Related Fields. 120. 183-208 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Setsuo Taniguchi : "Levy's stochastic area and the principle of stationary phase"Journal of functional Analysis. 172. 165-176 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Setsuo Taniguchi: "Analytic functions on abstract Wiener spaces"Journal of functional Analysis. (印刷中).

    • Related Report
      2000 Annual Research Report
  • [Publications] Hiroyuki Matsumoto,Naomasa Ueki: "Quadratic Hamiltonians, Wiener functionals and the metaplectic representations"Journal of Mathematical Society of Japan. 52. 269-292 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Catherine Donati-Martin,Hiroyuki Matsumoto,Marc Yor: "On positive and negative moments of the integral of geometric Brownian motions,"Statistics and Probability Letter. 49. 45-52 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Kumi Yasuda: "On Infinitely Divisible Distributions on Locally Compact Abelian Groups"Journal of Theoretical Probability. 13・3. 635-657 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Kumi Yasuda: "Extension of Measures to Infinite Dimensional Spaces over p-adic Field"Osaka Journal of Mathematics. 37. 967-985 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Tanigushi: "Stochastic oscillatory integrals with quadratic phase function and Jacobi equations"Probab.Theor.and Rel.Fields. 114・3. 291-308 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Sugita and S.Taniguchi: "A remark on stochastic oscillatory integrals with respect to a pinned wiener measure"Kyushu Jour.Math.. 53・2. 151-162 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Taniguchi: "Levy's stochastic area md the principle of stationamy phase"Jour.Funct.Anal.. (印刷中). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Kunita: "Analyticity md injectivity of convolution semigroups on Lie groups"Jour.Funct.Anal.. 165. 80-100 (1999)

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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