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

二阶Ventsel边界条件扩散过程的解析方法及其应用

• 批准号: 12640111
• 负责人: TSUCHIYA Masaaki
• 金额: $ 1.66万
• 依托单位: Kanazawa University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640111/
• 关键词: 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

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.

1. 我们用二阶文塞尔边界条件来处理扩散过程，通过构造相应的扩散方程的基本解来验证该过程的转移概率密度的存在；我们还证明了转移的严格正性。此外，利用转移概率密度，我们得到了扩散过程状态空间域边界上的局部时间势的显式公式。 2． C^∞ 平滑分数阶流形和 Holder 映射空间上的 Whitney 拓扑。 3. 接下来我们考虑与局部类型狄利克雷形式相关的马尔可夫过程的收敛性，而不假设极限过程的基本测度是非简并的。极限过程一般不是扩散，而近似过程是扩散，因此我们通过获得相应的带有边界条件的积分微分方程来对极限过程进行解析表征。这是与Y的共同工作。 Ogura和M. Tomisaki.4.最后，利用惩罚方法，考虑了具有反射边界条件的随机微分方程解的强近似，并结合实解析方法，得到了Paley等式和Hausdorff算子的结果。

项目成果

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), Birk

Yukio Ogura、Matsuyo Tomisaki 和 Masaaki Tsuchiya：“积分微分方程的强解的存在性和扩散过程的叠加”有限和无限维中的随机性。

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).

H. Kawakami，M. Tsuchiya：“分数阶流形的 C^∞ 平滑和霍尔德映射空间上惠特尼拓扑的基本属性”《国际应用数学杂志》（正在出版）。

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)

神奈川秀哉、小川茂吉：《随机微分方程的数值解法（应用版）》数学（日本数学会会刊）53. 125-138（2001）。

Yuichi Kanjin: "The Hausdorff operators on the real Hardy spaces HpR"Studia Mathematica. 148(1). 37-45 (2001)

Yuichi Kanjin：“真正的 Hardy 空间 HpR 上的 Hausdorff 算子”Studia Mathematica。

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)

Y.Ogura、M.Tomisaki、M.Tsuchiya：“积分微分方程的强解的存在和扩散过程的叠加”有限和无限维随机数学趋势，Birkhauser。