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ON THE DIFFUSION PROCESSES ON MOVING DOMAINS

Research Project

Project/Area Number 13640122
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 InstitutionKumamoto University

Principal Investigator

OSHIMA Yoichi  KUMAMOTO UNIV., ENG., PROF., 工学部, 教授 (20040404)

Co-Investigator(Kenkyū-buntansha) KIM Dae hong  KUMAMOTO UNIV., ENG., LECT., 工学部, 講師 (50336202)
HITSUDA Masuyuki  KUMAMOTO UNIV., SCI., PROF., 理学部, 教授 (50024237)
NAITO Koichirou  KUMAMOTO UNIV., ENG., PROF., 工学部, 教授 (10164104)
TAKEDA Masayoshi  TOHOKU UNIV., SCI., PROF., 理学研究科, 教授 (30179650)
Project Period (FY) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2002: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2001: ¥1,700,000 (Direct Cost: ¥1,700,000)
KeywordsSpace-time diffusion processes / Dirichlet forms / Reflecting processes / Local time / Skorohod representation / Optimal stopping problem / 時間依存ディリクレ形式 / 反射壁拡散過程 / 最適停止
Research Abstract

The main purpose of this research is to construct and give the properties of the diffusion processes on the domains which move depending on time. We assume that not only the domain but also the generator allow depending on time. The typical processes are absorbing or reflecting diffusion processes. In our case, difficulty arises because the boundary changes depending on time. In this research, after reorganizing the basic theory of time dependent Dirichlet forms, we constructed the processes on moving domains and get some properties of them. The fundamental settings are as follows. The domains are related by a time dependent maps of a fixed domain. The generator may depend upon the time. We first construct the diffusion processes on a fixed domain. For the construction, we use the general theory of time dependent Dirichlet forms. After that, we map the process to get the desired processes. In this argument, since the surface elements depend on time, to show that the mapped processes sa … More tisfy the desired property, we need the stochastic calculus including the Girsanov type transformations. Such stochastic calculus is given separately but not systematically in the time dependent case. Since we use such calculus in many places, the fundamental calculus will be published. As an important case, we consider the reflecting diffusion processes. Since the boundary changes depending on time, it is not easy to specify the local time and the normal derivative of the boundary. In this research, by using the map among the domains, we give the notions and define the local time on the boundary. Using the local time, we give the Skorohod representation of the diffusion processes. It is possible to apply our processes. One of the applications is an optimal stopping problem of time inhomogeneous diffusion processes. In the time inhomogeneous case, since there exist semipolar sets, it is not easy to characterize the optimal stopping time. For this problem, we can give the characterization of the time. We are considering that further related applications are possible. Less

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • Research Products

    (38 results)

All Other

All Publications (38 results)

  • [Publications] Y.Oshima: "On the exceptionality of some semipolar sets of time inhomogeneous Markov processes"Tohoku Math.J.. 54. 443-449 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Oshima: "On a construction of diffusion processes on moving domains"Potential Analysis. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Naito: "Recurrent dimensions of quasi-periodic orbits with irrational frequencies given by weak Liouville numbers"Proc.of International Conference on Mathematical Analysis and Applications. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Naito: "Recurrent dimensions of quasi-periodic solutions for non-linear evolution equations"Trans.Amer.Math.Soc.. 354. 1137-1151 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Naito: "Recurrent dimensions of quasi-periodic orbits with irrational frequencies given by quasi Liouville numbers"N onlinear Analysis. 47. 3671-3682 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Naito: "Recurrent dimensions of quasi-periodic orbits with irrational frequencies given by weak Liouville numbers"数理解析研究所講究録. 1187. 131-142 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Hibino, M.Hitsuda, H.Muraoka: "Remarks on a noncanonical representation for a stationary Gausian process"Acta Applicandae Mathematicae. 63. 137-139 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Hibino, M.Hitsuda: "Canonical propetry of representations of Gaussian processes with singular Volterra kernels"Infinite Dimensional Analysis, Quanturh Probability and Related Topics. 5. 293-296 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] D.Kim, M.Takeda, J.Ying: "Some variational formulas on additive functionals of symmetric Markov chains"Proc.Amer.Math.Soc.. 130. 2115-2123 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Takeda: "Conditional gaugeability and subcriticality of genegalized Schroedinger operators"J.Funct.Anal.. 191. 343-376 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Oshima: "On the exceptionality of some semipolar sets of time inhomogeneous Markov processes"Tohoku Math J.. 54. 443-449 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Oshima: "On a construction of diffusion processes on moving domains"Potential Analysis. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Naito: "Recurrent dimensions of quasi-periodic orbits with irrational frequencies given by weak Liouville numbers"Proc. International Conference on Mathematical Analysis and Applications. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Naito: "Recurrent dimensions of quasi-periodic solutions for non-linear evolution equations"Trans. Amer. Math. Soc.. 354. 1137-1151 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Naito: "Recurrent dimensions of quasi-periodic orbits with irrational frequencies given by quasi Liouville numbers"Nonlinear Analysis. 47. 3671-3682 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Naito: "Recurrent dimensions of quasi-periodic orbits with irrational frequencies given by weak Liouville numbers"1187. 131-142 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Hibino, M. Hitsuda, H. Muraoka: "Remarks on a noncanonical representation for a stationary Gaussian process"Acta Applicandae Mathematicae. 63. 137-139 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Hibino, M. Hitsuda: "Canonical property of representations of Gaussian processes with singular Volterra kernels"Infinite Dimensional Analysis, Quantum Probability and Related Topics. 5. 293-296 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] D. Kim, M. Takeda, J. Ying: "Some variational formulas on additive functionals of symmetric Markov chains"Proc. Amer. Math. Soc.. 130. 2115-2123 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Takeda: "Conditional gaugeability and subcriticality of generalized Schroedinger operators"J. Funct. Anal.. 191. 343-376 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Oshima: "On the exceptionality of some semipolar sets of time inhomogeneous Markov processes"Tohoku Math. J.. 54. 443-449 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Oshima: "On a Construction of Diffusion Processes on Moving Domains"Potential Analysis. (To appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] D.Kim: "Some variational formulas on additive functionals of symmetric Markov chains"Proc. Amer. Math. Soc.. 130. 2115-2123 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Takeda: "Conditional gaugeability and subcriticality of generalized Schroedinger operators"J. Funct. Anal.. 191. 343-376 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Hibino: "Canonical property of representations of Gaussian processes with singular Volterra kernels"Infinite Dimensional Analysis, Quantum Probability and Related Topics. 5. 293-296 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Naito: "Recurrent Dimensions of Quasi-Periodic Solutions for Nonlinear Evolution Equations II"Discrete and Continuous Dynamical Systems. (To appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Naito: "Recurrent dimensions of 3-frequencies quasi-periodic solutions for nonlinear evolution equations"Proc. Of the 1^<st> Korea-Japan Joint-seminar on Nonlinear Functional Analysis and Convex Analysis 2002. (To appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Naito: "Recurrent Dimensions of Quasi-Periodic Orbits with Frequencies given by Weak Liouville Numbers"International Journal of Mathematics, Game Theory and Algebra. (To appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Oshima: "On the exceptionality of some semipolar sets of time in homogeneous Markov processes"Tohoku Math. J.. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Naito: "Recurrent Dimensions of Quasi-Periodic Orbits with Irrational Frequencies given by Weak Liouville Numbers"Proc. of International Conference on Mathematical Analysis and Applications. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Naito: "Recurrent Dimensions of Quasi-Periodic Solutions for nonlinear evolution equations"Trans. Amer. Math. Soc.. 354・3. 1137-1151 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Naito: "Recurrent Dimensions of Quasi-Periodic Orbits with Irrational Frequencies given by Quasi Liouville Numbers"Nonlinear Analysis. 47. 3671-3682 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Naito: "Recurrent dimensions of quasi-periodic orbits with frequencies given by weak Liouville numbers"数理解析研究所講究録. 1187. 131-142 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Hibino, M.Hitsuda, H.Muraoka: "Remarks on a noncanonical representation for a stationary Gaussian process"Acta Applicandae Mathematicae. 63. 137-139 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Hibino, M.Hitsuda: "Canonical property of representations of Gaussian processes with singular Volterra kernels"Infinite Dimensional Analysis, Quantum Probability and Related Topics. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Takeda: "Conditional Gaugeability and Subcriticality of Genelalized Schrodinger Operators"J. Funct. Anal.. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] D.H.Kim, M.Takeda, J.Ying: "Some variational formulas on additive functionals of symmetric Markov chains"Proc. Amer. Math. Soc.. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] D.H.Kim: "Asymptotics of the spectral gaps related to energy forms on non-smooth domain"Potential Analysis. (to appear).

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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