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2005 Fiscal Year Final Research Report Summary

Research on phenomenon caused by simultaneous change of smooth measure and energy measure associated with Dirichlet forms

Research Project

Project/Area Number 15540121
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 InstitutionNara Women's University

Principal Investigator

TOMISAKI Matsuyo  Nara Women's University, Faculty of Sciences, Dept.of Math., Professor, 理学部, 教授 (50093977)

Co-Investigator(Kenkyū-buntansha) SHINODA Masato  Nara Women's University, Faculty of Sciences, Dept.of Math., Associated Professor, 理学部, 助教授 (50271044)
Project Period (FY) 2003 – 2005
KeywordsDirichlet forms / smooth measure / energy measure / distorted Brownian motion / generalized diffusion process / generalized diffusion operator / spectrum
Research Abstract

1.We treated phenomenon caused by simultaneous change of smooth measure and energy measure as a change of sequence of density functions of measures. When density functions of two measures coincide with each other and the parameter is transformed to a suitable one, a sequence of graphs of density function forms a family of functions whose length of graphs is bounded above. Therefore we can take a subsequence of density functions which converges to a limit function uniformly. In the case that the graph of the limit function is simple, we can find a division of a space of parameters. The division leads us to a family of one-dimensional generalized diffusion processes and a family of spectral measures. The spectral of limit process is represented by such family. In general, it is not easy to get such division of a space of parameters. We characterized a space of normalized measures, and hence we could control the behavior of measures. Thus we obtained a limit process.
2.For generalized diffusion processes with discrete spectrum, we showed that there exists a nontrivial limit distribution of conditional distributions related to hitting times. We obtained an asymptotic behavior of transition probability conditioned by no hitting to the boundaries as time goes to infinity. Our results say that the asymptotic behavior is affected by the asymptotic behavior of sample paths near the boundaries. Further the asymptotic behavior drastically changes according to discrete or continuous spectrum.
3.We defined models of percolation for Sierpinski carpet lattices and its family and showed that there exits a model for which there does not exist phase transition.

  • Research Products

    (6 results)

All 2003 Other

All Journal Article (6 results)

  • [Journal Article] A conditional limit theorem for generalized diffusion processes2003

    • Author(s)
      Z.Li
    • Journal Title

      Journal of Mathematics of Kyoto University 43・3

      Pages: 567・583

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Non-existence of phase transition of oriented percolation on Sierprinski carpet lattices2003

    • Author(s)
      M.Shinoda
    • Journal Title

      Probability Theory and Related Fields 125

      Pages: 447・456

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A conditional limit theorem for generalized diffusion processes2003

    • Author(s)
      Z.Li, T.Shiga, M.Tomisaki
    • Journal Title

      Journal off Mathematics of Kyoto University 43-3

      Pages: 567-583

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Non-existence of phase transition of oriented percolation on Sierpinski carpet lattices2003

    • Author(s)
      M.Shinoda
    • Journal Title

      Probability Theory and Related Fields 125

      Pages: 447-456

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Asymptotic conditional distributions related to one-dimensional generalized diffusion processes

    • Author(s)
      M.Iizuka
    • Journal Title

      Tsukuba Journal of Mathematics (掲載予定)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Asymptotic conditional distributions related to one-dimensional generalized diffusion processes

    • Author(s)
      M.Iizuka, M.Maeno, M.Tomisaki
    • Journal Title

      Tsukuba Journal of Mathematics (to appear in)

    • 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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