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An anlytic approach to diffusion processes with second order Ventsel's boundary conditions and its applications

Research Project

Project/Area Number 12640111
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 InstitutionKanazawa University

Principal Investigator

TSUCHIYA Masaaki  Kanazawa Univ. Faculty of Engineering ; Prof., 工学部, 教授 (50016101)

Co-Investigator(Kenkyū-buntansha) KAWAKAMI Hajime  Akita Univ. Fac. Engi. Resource Sci. ; Asso. Prof., 工学資源学部, 助教授 (20240781)
KANAGWA Syuya  Kanazawa Univ. Faculty of Engineering ; Prof., 工学部, 教授 (50185899)
KANJIN Yuichi  Kanazawa Univ. Faculty of Engineering ; Prof., 工学部, 教授 (50091674)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2001: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2000: ¥1,400,000 (Direct Cost: ¥1,400,000)
Keywordsdiffusion process / Ventsel's boundary condition / Dirichlet form / stochactic differential equation / penalty method / numerical analysis / Hausdorff operator / Paley's inequality / 基本解 / ペーリー不等式 / 微積分方程式 / 確率過程の強近インス / モンテカルロシミュレーション / 離散ハンケル空間 / アトム分解
Research Abstract

1. We treat diffusion processes with second order Ventsel's boundary conditions. The existence of a transition probability desity for such a process is verified ; it is done by constructing a fundamental solution of the corresponding diffusion equation. We also show the strict positivity of the transition probability density. Furthermore, using the transition probability density, we obtained an explicit formula for the potential of the local time on the boundary of the domain which is the state space of the diffusion process.
2. To get the result mentioned above, we study C^∞ smoothing of manifolds with fractional order and the Whitney topology on the spaces of Holder maps.
3. Next we consider the convergence of Markov processes associated with local type Dirichlet forms without assuming that the basic measure of the limit process is non- degenerate. In this case, the limit process is not diffusion in general, whereas the approximate processes are diffusion. Hence we give an analytic chacterization for the limit process by obtaining the corresponding integro-differential equation with boundary condition. This is a joint work with Y. Ogura and M. Tomisaki.
4. Finally, using penalty method, strong aproximation to the solutions of stochastic differential equations with reflecting boundary condition is considered. In connection with real analytic approach, we obtain results on Paley's iequality and Hausdorff operator.

Report

(3 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • Research Products

    (26 results)

All Other

All Publications (26 results)

  • [Publications] Y.Ogura, M.Tomisaki, M.Tsuchiya: "Existence of a strong solution for an integro-differential equation and superposition of diffusion processes"Stochastic in Finite and Infinite Dimensions Trends in Mathematics, Birkhauser. 341-359 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Kawakami, M.Tsuchiya: "C^∞ smoothing of manifolds of fractional order and basic properties of the Whitney topology on the spaces of Holder maps"Internationa Journal of Applied Mathematics. 6. 319-340 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y.Ogura, M.Tomisaki, M.Tsuchiya: "Convergence of local type Dirichlet forms to a non-local type one"Annales de I'Institut Henri Poincare ; Probabilites et Statistiques. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y/Kanjin, K.Sato: "Paley's inequality for the Jacobi expansions"Bulletin of the London Mathematical Society. 33. 483-491 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y.Kanjin: "The Hausdorff operators on the real Hardy spaces Hp(R)"Studia Mathematica. 148. 37-45 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Kanagawa, Y.Saisho: "Strong approximation of reflecting Brownian motion using penalty method and its application to computer simulation"Monte Carlo Methods and Applications. 6. 105-114 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 金川秀也, 小川重義: "確率微分方程式の数値解法(応用編)"数学(日本数学会機関誌). 53. 125-138 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Kanagawa: "Some remarks on strong approximation of reflecting Brownian motion using penalty method"無限分解可能分布に関連する諸問題(6) 統計数理研究所共同研究リポート. 146. 26-32 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yukio Ogura, Matsuyo Tomisaki, and Masaaki Tsuchiya: "Existence of a strong solution for an integro-differential equation and superposition of diffusion processes"Stochastic in Finite and Infinite Dim ensi ons . Trends in Mathematics, 341-359 (2001), Birkhauser.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hajime Kawakami, and Masaaki Tsuchiya: "C^∞ smoothing of manifolds of fractional order and basic properties of the Whitney topology on the spaces of Holder maps"International Journal of Applied Mathematics. 6(3). 319-340 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yukio Ogura, Matsuyo Tomisaki, and Masaaki Tsuchiya: "Convergence of local type Dirichlet forms to a non-local type one"Annales de 1' Institut Henri Poincare ; Probabilites et Statistiques. (In Press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yuichi Kanjin, and Kunio Sato: "Paley's inequality for the Jacobi expansions"Bulletin of the London Mathematical Society. 33(4). 483-491 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Yuichi Kanjin: "The Hausdorff operators on the real Hardy spaces HpR"Studia Mathematica. 148(1). 37-45 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Shuya Kanagawa, and Yasumasa Saisho: "Strong approximation of reflecting Brownian motion using penalty method and its application to computer simulation"Monte Carlo Methods and Applications. 6(2). 105-114 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Shuya Kanagawa, and Shigeyoshi Ogawa: "Numerical analysis to stochastic differential equations (Part of Application)"Sugaku (In Japanese). 53(2). 125-138 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Syuya Kanagawa: "Some remarks on strong approximation of reflecting Brownian motion using penalty method"Infinitely Divisible Processes and Related Topics (6) The Institute of Statistical , Mathematics Cooperative Research Report. 146. 26-32 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y.Ogura, M.Tomisaki, M.Tsuchiya: "Convergence of local type Dirichlet forms to a non-local type one"Annales de L'Institut Henri Poincare ; Probabilites et Statistiques. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Kawakami. M.tsuchiya: "C^∞ smoothing of manifolds of fractional order and basic properties on the Whitney topology of the Holder maps"International Journal of Applied Mathematics. 6・3. 319-340 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Kanjin: "The Hausdorff operators on the real Hardy spaces Hp(R)"Studia Mathematica. 148・1. 37-45 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.kanjin, K.Sato: "Paley's inequality for the Jacobi expansions"Bulletin of the London Mathematical Society. 33・4. 483-491 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 金川秀也, 小川重義: "確率微分方程式の数値解法(応用編)"数学(日本数学会機関紙). 53・2. 125-138 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Ogura,M.Tomisaki,M.Tsuchiya: "Existence of a strong solution for an integro-differential equation and superposition of diffusion processes,"Stochastics in Finite and Infinite Dimensions (Trends in Mathematics) Birkhauser. 341-359 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Kawakami,M.Tsuchiya: "C^∞ smoothing of manifolds of fractional order and basic properties of the Whitney topology on the spaces of Holder maps"International Journal of Applied Mathematics. (印刷中).

    • Related Report
      2000 Annual Research Report
  • [Publications] Y.Kanjin,M.Satake: "Inequalities for discrete Hardy spaces"Acta Mathematica Hungarica. 89・4. 301-313 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Kanagawa,Y.Saisho: "Strong approximation of reflecting Brownian motion using penalty method and its application to computer simulation"Monte Carlo Methods and Applications. 6. 105-114 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 小川重義,金川秀也: "確率微分方程式の数値解法2,応用編"日本数学会「数学」論説. (出版予定).

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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