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2002 Fiscal Year Final Research Report Summary

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
KeywordsSpace-time diffusion processes / Dirichlet forms / Reflecting processes / Local time / Skorohod representation / Optimal stopping problem
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

  • Research Products

    (20 results)

All Other

All Publications (20 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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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

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Published: 2004-04-14  

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