Study of path-behaviors of Biharmonic Psuedo Process with boundaries

有边界双调和伪过程的路径行为研究

• 批准号: 10640128
• 负责人: NISHIOKA Kunio
• 金额: $ 1.92万
• 依托单位: TOKYO METOROPOLITAN UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640128/
• 关键词: biharmonic operator; monopole; dipole; absorbing boundary; sticky boundary; reflecting boundary; Poisson分布; 無限次元力学系; 弱い意味での確率積分; Girsanovの公式; 4階非線形偏微分方程式; 確率解; カオス

-△ィイD12ィエD1 is called a biharamonic operator, which we often encounter in the theory of elasticity and fluid dynamics. Following pioneer works of Hochberg (1978) and Funaki (1979), many attempts are done to study the operator from the view point of probability theory.We consider 'particles' whose 'transition probability density' is the fundamental solution of ∂ィイD2tィエD2u = -△ィイD12ィエD1u, though it may have negative values. We call orbits of these 'particles' biharmonic pseudo process, or BPP in short. Our aim is to study relations between analytic boundary conditions of -△ィイD12ィエD1 and behaviors of BPP.By Nishioka (1997), we know that BPP behave as a mixture of two sorts of particles which we call 'monopoles' and 'dipoles'. So we need two boundary conditions to solve a boundary value problem related to -△ィイD12ィエD1, since each boundary condition controls behavior of each sort of particles at the boundary. We set up one of 'absorbing', 'sticking', and various 'reflecting' barriers on monopoles and dipoles one by one. Then their combinations solve various kinds of boundary value problems to -△ィイD12ィエD1.

-II II D12E D1被称为Biharamonic操作员，我们经常在弹性和流体动力学理论中遇到。遵循Hochberg（1978）和Funaki（1979）的开拓者作品，从概率理论的角度研究了操作员进行了许多尝试。我们考虑其“颗粒”的“过渡概率密度”是∂iiid2te d2te d2te D2U = -YII II D12E D1U的基本解决方案，尽管它可能具有负值。我们称这些“颗粒”的轨道为轨道，或者简称为BPP。我们的目的是研究-II D12E D1的分析边界条件与BPP的行为之间的关系。ByNishioka（1997），我们知道BPP的行为是我们称为“单管”和“偶极子”的两种粒子的混合物。因此，我们需要两个边界条件来解决与-II D12E D1相关的边界值问题，因为每个边界条件都控制边界处每种粒子的行为。我们逐一设置了一种“吸收”，“粘住”和各种“反射”障碍物。然后，他们的组合将各种边界价值问题求解为-II D12。

项目成果

岡部恒春,西岡國雄,et al: "数学英和小辞典"講談社サイエンティフィック. 294 (1999)

Tsuneharu Okabe、Kunio Nishioka 等：《数学英日词典》讲谈社科学 294 (1999)。

K.Nishioka: "The first hitting time and place of a half-line by a biharmonic pseudo process" Japan.J.Math.23. 235-280 (1997)

K.Nishioka：“通过双调和伪过程首次击球半线的时间和地点”Japan.J.Math.23。

H. Hirata, B. Sassol, S. Vaienti: "Statistics of Return Times : A General Framework and New Applications"Comm. Math. Phys.. 206. 33-55 (1999)

H. Hirata、B. Sassol、S. Vaienti：“返回时间统计：一般框架和新应用”Comm。

M. Hirata (with B. Saussol and S. Vaienti): "Statistics of return times: A general frame work and new application"Comm. Math. Phys.. 206. 33-35 (1999)

M. Hirata（与 B. Saussol 和 S. Vaienti）：“返回时间统计：一般框架和新应用”Comm。

N. Aoki: "Differentiable maps having hyperbolic sets"Topology Appl.. 82. 15-48 (1998)

N. Aoki：“具有双曲集的可微映射”拓扑应用 82. 15-48 (1998)