| Project/Area Number |
12640111
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kanazawa University |
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Principal Investigator |
TSUCHIYA Masaaki Kanazawa Univ. Faculty of Engineering ; Prof., 工学部, 教授 (50016101)
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| 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)
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| Project Period (FY) |
2000 – 2001
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| Project Status |
Completed (Fiscal Year 2001)
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| 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)
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| Keywords | diffusion 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 |
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| 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.
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