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2005 Fiscal Year Final Research Report Summary

Large deviations for symmetric Markov processes and Dirichlet forms

Research Project

Project/Area Number 15540103
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionTohoku University

Principal Investigator

TAKEDA Masayoshi  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (30179650)

Co-Investigator(Kenkyū-buntansha) HATTORI Tetsuya  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10180902)
SHIOYA Takashi  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90235507)
Project Period (FY) 2003 – 2005
KeywordsMarkov process / Dirichlet form / Feynman-Kac functional / time-changed process
Research Abstract

We proved that the integrability of Feynman-Kac functionals (gaugeability) is equivalent to that the principal eigenvalue of time-changed process is greater than 1, which is also equivalent to the subcriticality of Schroedinger operators. This fact says that the principal eigenvalue of time-changed process accurately measures the size of measures. Using this fact, we obtained three results : The first result is that the ultracontractiyity of Schroedinger semigroups holds if and only if the princilal eigenvalue of time-changed process is greater than 1. The second result is that the expectation of the number of branches hitting a closed set in a branching symmetric stable process is finite if and only if the princilal eigenvalue is greater than 1. The final result is as foolows : Suppose that the heat kernel on a complete Riemannian manifold satisfies the global Gaussian bounds, so called Li-Yau estimate. Then the heat kernel of the Schroedinger operator also possesses the global Gaussian bounds, if and anly if the princilal eigenvalue is greater than 1.

  • Research Products

    (12 results)

All 2005 2004 Other

All Journal Article (12 results)

  • [Journal Article] Variational formula for Dirichlet forms and estimates of principal eigenvalues for symmetric α-stable processes2005

    • Author(s)
      竹田雅好
    • Journal Title

      Potential Analysis 23

      Pages: 135-151

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Displacement exponents of self-repelling walks and self-attracting walks on the pre-Sierpinski gasket2005

    • Author(s)
      服部哲弥
    • Journal Title

      Journal of Math. Sci. University of Tokyo 12

      Pages: 417-443

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Volume collapsed three-manifolds with a lower curvature bound2005

    • Author(s)
      塩谷隆
    • Journal Title

      Math. Ann. 333

      Pages: 131-155

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Variational formula for Dirichlet forms and estimates of principal eigenvalues for symmetric α-stable processes2005

    • Author(s)
      M.Takeda
    • Journal Title

      Potential Analysis 23

      Pages: 135-151

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Displacement exponents of self-repelling walks and self-attracting walks on the pre-Sierpinski gasket2005

    • Author(s)
      T.Hattori
    • Journal Title

      Journal of Math.Sci. University of Tokyo 12

      Pages: 417-443

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Volume collapsed three-manifolds with a lower curvature bound2005

    • Author(s)
      T.Shioya
    • Journal Title

      Math.Ann. 333

      Pages: 131-155

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Subcriticality and gaugeability for symmetric α-stable processes2004

    • Author(s)
      竹田雅好
    • Journal Title

      Forum math. 16

      Pages: 505-517

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Subcriticality and gaugeability for symmetric α-stable processes2004

    • Author(s)
      M.Takeda
    • Journal Title

      Forum math. 16

      Pages: 505-517

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Differentiability of spectral functions for symmetric α-stable processes

    • Author(s)
      竹田雅好
    • Journal Title

      Trans. Amer. Math. Soc. (In press)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Gaugeability for Feynman-Kac functionals with applications to symmetric α-stable processes

    • Author(s)
      竹田雅好
    • Journal Title

      Proc. Amer. Math. Soc. (In press)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Differentiability of spectral functions for symmetric α-stable processes

    • Author(s)
      M.Takeda
    • Journal Title

      Trans.Amer.Math.Soc. (In press)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Gaugeability for Feynman-Kac functionals with applications to symmetric α-stable processes

    • Author(s)
      M.Takeda
    • Journal Title

      Proc.Amer.Math.Soc. (In press)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2007-12-13  

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