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Microlocal analysis and pseudo-differential operators

Research Project

Project/Area Number 14340045
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

KOTANI Shinichi (2003-2005)  Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10025463)

長瀬 道弘 (2002)  大阪大学, 大学院・理学研究科, 教授 (70034733)

Co-Investigator(Kenkyū-buntansha) MANABE Shojiro  Osaka University, Kyouikujissenn Center, Professor, 大学教育実践センター, 教授 (20028260)
SUGIMOTO Mituru  Osaka University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (60196756)
UCHIDA Motoo  Osaka University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (10221805)
FURIHATA Daisuke  Osaka University, CyberMedia Center, Associate Professor, サイバーメディアセンター, 助教授 (80242014)
ISOZAKI Yasuki  Osaka University, Graduate School of Science, Lecturer, 大学院・理学研究科, 講師 (90273573)
芦野 隆一  大阪教育大学, 教育学部, 助教授 (80249490)
岩崎 千里  兵庫県立大学, 大学院・物質理学研究科, 教授 (30028261)
小谷 眞一  大阪大学, 大学院・理学研究科, 教授 (10025463)
Project Period (FY) 2002 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥11,400,000 (Direct Cost: ¥11,400,000)
Fiscal Year 2005: ¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2004: ¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2003: ¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2002: ¥4,400,000 (Direct Cost: ¥4,400,000)
Keywordsdispersive equation / index theorem / smoothing property / Wavelet / Burgers equation / Schroedinger equation / KdV equation / Krein's theory / 超局所解析 / 擬微分作用素 / マルチンゲール / シュレーディンガー作用素 / スペクトル / 超関数 / 波面集合 / L^p(R^n)-有界性 / 表象 / 磁場ポテンシャル / 量子化
Research Abstract

1.Differential equations (Uchida, Iwasaki, Sugimoto)
(1)The local index theorem was extended to manifolds with boundary and Koeler manifolds by applying heat equations.
(2)Hoermander's uniqueness theorem was proved geometrically by formulating his strong pseudo-convex condition for general differential equation systems.
(3)The smoothing property for dispersive equations was considered by transforming by simpler systems through associated dynamical systems.
2.Numerical analysis and applications of pseudo-differential operators (Ashino)
(1)Decomposition of singular values and wavelet analysis were applied to image analysis.
(2)Wavelet was applied to identify the systems.
3.Probability theory (Kotani, Isozaki)
(1)By extending the deviation theorem, a theorem including three random variables was obtained and was applied to the investigation of the distribution of the hitting time to the half line of 2-dim. random walks.
(2)A final answer for the distribution of clusters created by Burgers equations with random initial values was obtained.
(3)Higher dimensional Schroedinger operators with random potentials was studied by introducing an exponents of the Green functions corresponding to the Lyapounov exponents in 1-dim.. For 1-dim. operators, the KdV-flow was constructed by employing Sato's theory.
(4)1-dim. diffusion processes were studied from the point of view of martingales and Krein's spectral theory.

Report

(5 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (32 results)

All 2006 2005 2004 2003 Other

All Journal Article (20 results) Publications (12 results)

  • [Journal Article] On some laws of iterated logarithm for Burgers turbulence with Brownian initial data based on the concave majorant.2006

    • Author(s)
      Y.Isozaki
    • Journal Title

      Osaka J.Math., Vol43 no.2(in press)

    • NAID

      120004845906

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Global calculus of Fourier integral operators, weighted estimates, and applications to global analysis of hyperbolic equations2006

    • Author(s)
      M.Sugimoto
    • Journal Title

      Operator Theory : Advances and Applications, Birkhauser Verlag, 164

      Pages: 65-78

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Global smoothing estimates for dispersive equations with non-polynomial symbols2005

    • Author(s)
      M.Sugimoto
    • Journal Title

      Proc. 12th International Conf. on Finite of Infinite Dim. Complex Anal. and Appl.

      Pages: 283-290

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Global smoothing estimates for dispersive equations with non-polynomial symbols2005

    • Author(s)
      M.Sugimoto
    • Journal Title

      Proc.12th International Conf.on Finite of Infinite Dim.Complex Anal.and Appl.

      Pages: 283-290

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A smoothing property of Schrodinger equations and a global existence result for derivative nonlinear equations.,2005

    • Author(s)
      M.Sugimoto
    • Journal Title

      Advances in Analysis, World Sci.Publ., Hackensack, NJ

      Pages: 315-320

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Global smoothing estimates for dispersive equations with non-polynomial symbols,2005

    • Author(s)
      M.Sugimoto
    • Journal Title

      Proceedings of The 12th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications

      Pages: 283-290

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A new proof of global smoothing estimates for dispersive equations2004

    • Author(s)
      M.Sugimoto
    • Journal Title

      Advances in Pseudo-Differential Operators 155

      Pages: 65-75

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a condition that one-dimensional diffusion processes are martingales2004

    • Author(s)
      小谷眞一
    • Journal Title

      Seminaire de Probabilites (to appear)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A New proof of global smoothing estimates for dispersive equations2004

    • Author(s)
      杉本充, M.Ruzhansky
    • Journal Title

      Advances in Pseudo-Differential Operators, Birkhauser Basel 155

      Pages: 65-75

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Hoermander form and uniequenwss for the Cauchy problem2004

    • Author(s)
      内田素夫
    • Journal Title

      Advances in Mathematics 189

      Pages: 237-245

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A family of symmetric stable-like processes and its global path properties2004

    • Author(s)
      磯崎泰樹, T.Uemura
    • Journal Title

      Probab.Math.Statist 24

      Pages: 145-164

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Applications of wavelet transforms to system identification.2004

    • Author(s)
      芦野隆一, Mandai Takeshi, Morimoto Akira
    • Journal Title

      Advances in pseudo-differential operators, Oper.Theory Adv.Appl, Birkhauser, Basel 155

      Pages: 203-218

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Thermodynamic limit of relative Fredholm determinant of Green operators for random Schrodinger operators2003

    • Author(s)
      S.Kotani
    • Journal Title

      Markov Process & Related Fields 9

      Pages: 709-716

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Smooth tight frame wavelets and image microanalysis in the Fourier domain2003

    • Author(s)
      R.Ashino
    • Journal Title

      Computer Math. Applc 45/10-11

      Pages: 1551-1579

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Wavelet bascs for microlocal filtering and the sampling theorem in I!(R")2003

    • Author(s)
      R.Ashino
    • Journal Title

      Appl. Anal. 82/1

      Pages: 1-24

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Smooth tight frame wavelets and image microanalysis in the Fourier domain2003

    • Author(s)
      R.Ashino et al.
    • Journal Title

      Computer Math.Applc. 45/10-11

      Pages: 1551-1579

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Wavelet bases for microlocal filtering and the sampling theorem in L"(R")2003

    • Author(s)
      R.Ashino et al.
    • Journal Title

      Appl.Anal. 82/1

      Pages: 1-24

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a condition that one-dimensional diffusion processes are martingales

    • Author(s)
      S.Kotani
    • Journal Title

      Seminaire de Probabilites (in press)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a condition that one-dimensional diffusion processes are martingales

    • Author(s)
      S.Kotani
    • Journal Title

      Seminaire de Probabilites (in press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a condition that one-dimensional diffusion processes are martingales

    • Author(s)
      S.Kotani
    • Journal Title

      Seminaire de Probabilites (to appear)

    • Related Report
      2005 Annual Research Report
  • [Publications] S.Kotani: "Thermodynamic limit of relative Fredholm determinant of Green operators for random Schrodinger operators"Markov Process. Related Fields. (発表予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Uchida: "Hormander form and uniqueness for the Cauchy problem"J our Adv. in Math.. (発表予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] R.Ashino, C.Heil, M.Nagase, R.Vaillancourt: "Smooth tight frame wavelets and image microanalysis in the Fourier domain"Computer Math.Applc.. 45/10-11. 1551-1579 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] R.Ashino, T.Mandai: "Wavelet bases for microlocal filtering and the sampling theorem in L_p(R^n)"Appl.Anal. 82/1. 1-24 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Sugimoto: "A Smoothing property of Schrodinger equations along the sphere"J.Anal.Math.. 89. 15-30 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Isozaki, T.Uemura: "A family of symmetric stable-like processes and its global path properties"Probability and Mathematical Statistics. 24/1(To appear). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] 長瀬道弘, 芦野隆一, C.Heil: "Multiwavelets, pseudodifferential operators and microlocal analysis"AMS/IP Stud. Adv. Math.. 25. 9-20 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 長瀬道弘, 芦野隆一, C.Heil: "Microlocal analysis and multiwavelets"World Sci. Publishing. 293-302 (2000)

    • Related Report
      2002 Annual Research Report
  • [Publications] 杉本 充: "A weak extension theorem for inhomogeneous differential equations"Forum Math.. 13. 323-334 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] 内田 素夫: "A non-existence theorem of lacunas for hyperbolic differential operators with constant coefficients"Ark. Mat.. 40. 201-205 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 内田 素夫: "Regularity of weak solutions of semilinear elliptic differential operators"J. Math. Sci. Univ. Tokyo. 8. 357-363 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] 楳田登美男: "Eigenfunction expansions associated with relatiristic schrodinger operators"Oper. Theory Adv. Appl.. 126. 315-319 (2001)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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