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Research on differential operators in infinite dimensional spaces with symmetry

Research Project

Project/Area Number 06452014
Research Category

Grant-in-Aid for General Scientific Research (B)

Allocation TypeSingle-year Grants
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionThe University of Tokyo

Principal Investigator

KUSUOKA Shigeo  Professor, Graduate School of Mathematical Sciences, The University of Tokyo, 大学院・数理科学研究科, 教授 (00114463)

Co-Investigator(Kenkyū-buntansha) TSUTSUMI Yoshio  Assistant Professor, Graduate School of Mathematical Sciences, The University of, 大学院・数理科学研究科, 助教授 (10180027)
OSADA Hirofumi  Assistant Professor, Graduate School of Mathematical Sciences, The University of, 大学院・数理科学研究科, 助教授 (20177207)
KATAOKA Kiyoomi  Professor, Graduate School of Mathematical Sciences, The University of Tokyo, 大学院・数理科学研究科, 教授 (60107688)
MATSUMOTO Yukio  Professor, Graduate School of Mathematical Sciences, The University of Tokyo, 大学院・数理科学研究科, 教授 (20011637)
MATANO Hiroshi  Professor, Graduate School of Mathematical Sciences, The University of Tokyo, 大学院・数理科学研究科, 教授 (40126165)
Project Period (FY) 1994 – 1995
Project Status Completed (Fiscal Year 1995)
Budget Amount *help
¥5,300,000 (Direct Cost: ¥5,300,000)
Fiscal Year 1995: ¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 1994: ¥2,900,000 (Direct Cost: ¥2,900,000)
KeywordsWiener Measure / Stochastic Analysis / Skelton / Dolbealt Cohomology / Horomorphic Function / Infinite Dimensional Analysis / Complex Wiener Space / ウィナー多様体 / 無限次元空間 / 対称性 / ドルボーコホモロジー
Research Abstract

The object of this research was differential operators in infinite dimensional spaces, especially ones with symmetry. We had a progess in the research on holomorphic functions on complex Wiener spaces from viewpoint of stochastic analysis and in the research on asymptotics of fundamental solutions of heat equations with boundary conditions.
There have been some works on holomorphic functions in complex Wiener spaces. But all of them handled holomorphic functions defined in the whole space only. So these works did not make clear the holomorphicity as a local property. Since the aim of this research is to establish analysis on Wiener manifolds, we really need the localization of the concepts. We had a big progress in the localization of the notion of holomorphicity and skelton. The skelton of holomorphic functions is holomorphic functions in the Cameron-Martin space associated the original holomorphic functions. The existence of the skelton for holomorphic functions defined on the whole c … More omplex Wiener space was shown by Prof.Sugita in Kyushu University. But nothing is known about the skelton for holomorphic functions defined in subdomains. In this research we showed that if the subdomain is given by a positive domain of a good function, then there exists the skelton of holomorphic functions defined in the domain and the corespondance of skelton and holomorphic functions is one-to-one. Combining this result with known results, we can see that the Dolbeault cohomology coincides with the Cech cohomology of the sheaf of holomorphic functions in the domain. We hope that this result plays an important role in complex analysis in infinite dimensional spaces.
We also obtained the precise estimates on the asymptotic behavior of the fundamental solution for the heat equation with Dirichlet or Neumann conditions in the outside of strictly convex domains with smooth boundaries. We obtained this result by representing the fundamental solutions by the Wiener measure and by using the symmetry. We did not expect such a result, but we obtained it as a by-product. Less

Report

(3 results)
  • 1995 Annual Research Report   Final Research Report Summary
  • 1994 Annual Research Report
  • Research Products

    (27 results)

All Other

All Publications (27 results)

  • [Publications] S. Kusuoka: "A basic estimate for two-dimensional holonomy" Journal of Functional Analysis. 127. 132-154 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] S. Kusuoka: "Limit theorem on option replication with transaction costs" Annals of Applied Probability. 5. 198-221 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] S. Kusuoka(N. Ikeda): "Short time asymptotics for fundamental solutions of heat equations" Proc. Taniguchi Conf.(発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] H. Osada: "An invariance principle for non-symmetric Markor process and reflecting diffusions in random domains" Prob. Theory and Related Fields. 101. 45-63 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] H. Osada: "Self-similar diffusions on a class of infinite ramified fractals" J. of Mathematical Society Japan. 47. 591-616 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] Y. Tsutsumi: "Global existence and uniqueness of energy solutions for the Maxwell Srodinger equations" Hokkaido Mathematical Journal. 24. 617-639 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] 楠岡成雄: "確率・統計" 森北出版, 102 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] S.KUSUOKA: "A basic estimate for two-dimensional holonomy" Journal of Functional Analysis. 127. 132-154 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] S.KUSUOKA: "Limit theorem on option replication with transaction costs" Annals of Applied Probability. 5. 198-221 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] N.IKEDA and S.KUSUOKA: "Short time asymptotics for fundamental solutions of heat equations" Proceedings of Taniguchi Conference. (to appear). (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] H.OSADA: "An invariance principle for non-symmetric Markov process and reflectig diffusions in random domains" Probability Theory and Related Fields. 101. 45-63 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] H.OSADA: "Selp-similar diffusions on a class of infinite ramified farctals" Journal of Mathematical Society of Japan. 47. 591-616 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] Y.TSUTSUMI: "Global existence and uniqueness of energy solutions for the Maxwell Scrodinger equations" Hokkaido Mathematical Journal. 24. 617-639 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1995 Final Research Report Summary
  • [Publications] 楠岡成雄: "Limit theorem on option replication with transaction costs" Annals of Applied Probability. 5. 198-221 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] 楠岡成雄(池田信行): "Short time asymptotics for fundamental solutions of heat equations" Proc.Taniguchi Conf.(発表予定).

    • Related Report
      1995 Annual Research Report
  • [Publications] 長田博文: "An invariance principle for non-symmetric Markor processes and reflecting diffusions in random domains" Prob.Theory and Related Fields. 101. 45-63 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] 長田博文: "Self-similar diffusions on a class of infinited ramified fractals" Journal of Mathematical Society Japan. 47. 591-616 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] 提誉志雄: "Global existence and uniqueness of enegy solutions for the Maxwell-Schrodinger equations" Hokkaido Mathematical Journal. 24. 617-639 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] 提誉志雄(小沢徹): "Normal form and global solutions for the Klein-Gordon-Zakharov equations" Annales de I'Institut Henri Poincare. 12. 459-503 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] 楠岡成雄: "確率・統計" 森北出版, 102 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] S.Kusuoka: "Limit theorem on option replication with transaction costs" Annals of Applied Probability. 5(発表予定). (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] S.Kusuoka: "A basic estimate for twodimensinal stochastic holonomy" Journal of Functional Analysis. 127. 132-154 (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] Y.Matsumoto: "On the topological structure of the Fermat surface of degree 5" Kodai Math.J.17. 560-570 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] Y.Tsutsumi: "The null gauge condition and the one dimensional nonlinear Schrodinger equation with cubic nonlinearity" Indiana Univ.Math.J.43. 241-254 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] C.M.Elliott,H.Matano and T.Qi: "Zeros of a complex Ginzburg-Landau order parameter with applications to superconductivity" European Journal of Applied Mathematics. 5. 431-448 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] H.Osada: "An invariance principle for non-symmetric Markov processes and reflecting diffusions in random domains" Probab.Theory Relat.Fields. 101. 45-63 (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] 楠岡成雄: "確率と統計" 森北出版, 100 (1995)

    • Related Report
      1994 Annual Research Report

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Published: 1994-04-01   Modified: 2016-04-21  

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