• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to project page

1999 Fiscal Year Final Research Report Summary

Symmetric Markov processes and Dirichlet forms

Research Project

Project/Area Number 09640265
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 (1998-1999)
Osaka University (1997)

Principal Investigator

TAKEDA Masayoshi  Mathematics, Tohoku Univ., Professor, 大学院・理学研究科, 教授 (30179650)

Co-Investigator(Kenkyū-buntansha) TACHIZAWA Kazuya  Mathematics, Tohoku Univ., Lecturer, 大学院・理学研究科, 講師 (80227090)
IGARI Satoru  Mathematics, Tohoku Univ., Professor, 大学院・理学研究科, 教授 (50004289)
TAKAGI Izumi  Mathematics, Tohoku Univ., Professor, 大学院・理学研究科, 教授 (40154744)
NAGAI Hideo  Mathematical Science, Osaka Univ., Professor, 大学院・基礎工学研究科, 教授 (70110848)
Project Period (FY) 1997 – 1999
Keywordssymmetric Markov process / Dirichlet form / large deviation / additive functional / Feynman-Kac formula
Research Abstract

The objective of this study is to investigate symmetric Markov processes by using Dirichlet form theory. Symmetric Markov processes are a special class in Donsker-Varadhan type large deviation theory in the sense that the rate functions of large deviation principle are given by the associated Dirichlet forms. In 1984, Fukushima and I showed that symmetric Markov processes can be transformed to ergodic processes by some supermartingale multiplicative functionals even if a symmetric Markov process is explosive or has the killing inside. As a result, Donsker-Varadhan type large deviation principle could be extended to symmetric Markov processes with finite lifetime. In this study, I found a new sufficient condition for the upper estimate holding for not only compact sets but also for closed sets. In fact, I showed that the full large deviation principle holds if the Markov process explodes so fast that the 1-resolvent of the identity function belongs to the space of continuous functions vanishing at infinity. As a corollary of this result, I showed LィイD1pィエD1-independence of the spectral radius of symmetric Markov semigroups. And I applied it to obtain a necessary and sufficient condition for the integrability of Feynman-Kac functionals. This result also gives us an criterion whether a Schrodinger operators is subcritical or not.
We further extended the large deviation principle to Markov processes with Feynman-Kac functional, and consider asymptotic properties of Feynman-Kac semigroups.

  • Research Products

    (10 results)

All Other

All Publications (10 results)

  • [Publications] 竹田雅好: "A large deviation for symmetric Markor processes with finite life time"Stochastics and Stochastics Rep.. 59. 143-167 (1996)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 竹田雅好: "A symptotic propertions of additive functionals of zero energy"Ann.Probab.. 25. 940-952 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 竹田雅好: "Asymptotic properties of generalized Feymann-Kac functionals"Potential Analysis. 8. 261-291 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 竹田雅好: "Exponential decay of lifetime and a theorem of Kacon total occupation times"Potential Analysis. 11. 235-247 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 竹田雅好: "Large deviations and LIL's for Brownian motions on nested fractals"to appear in Osaka J. Math..

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M. Takeda: "A large deviation for symmetric Markov processes with finite lifetime"Stochastics and Stochastic report. 59. 143-167 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. Takeda: "Asymptotic properties of additive functionals of zero energy"Ann. Probab.. 25. 940-952 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. Takeda: "Asymptotic properties of generalized Feynman-Kac functionals"Potential Analysis. 8. 261-291 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. Takeda: "Exponential decay of lifetime and a theorem of Kac on total occupation times"Potential Analysis. 11. 235-247 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. Takeda: "Large deviations and LIL's for Brownian motions on Nested fractals"Osaka J. Math.. (to appear).

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

URL: 

Published: 2001-10-23  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi