Stochastic Analysis and Asymptotics

随机分析和渐进分析

• 批准号: 08454046
• 负责人: IKEDA Nobuyuki
• 金额: $ 2.05万
• 依托单位: Ritsumeikan University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1996
• 资助国家: 日本
• 起止时间: 1996 至 无数据
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08454046/
• 关键词: stochastic analysis; Wiener integral; path integral; Van Vleck formula; Jacobi field; local time; Selberg trance formula; Wiener測度

1. The Van Vleck formula for Feynman's path integrals with qusdratic phase functionals plays an important role in the study of the mathematical science. By using the method of stochastic analysis, N.Ikeda proved an analogous formula for Weiner integrals with quadratic phase functionals to the Van Vleck formula mentioned above. By using transformations described in terms of Jacpbi fields, we can rewrite quadratic Wiener functionals associated with quadratic Lagrangians in the canonical form on the Wiener space. Then the functional determinants appeared in the Van Vleck formula can be regarded as the Jacobian of the transformation above as in case of finite dimensional manifolds.2. Without using the exact formula of the fundamental solution of the heat equation on the Poincare upper half plane, N.Ikeda obtained a proof of Selberg trace formula based on the integral representation of the fundamental solution of the heat equation on the Wiener space.3. T.Yamada has studied principal values of Brownian local times and representations of Brownian additive functionals of zero energy. M.Arai obtained remarkable results about the growth order of eigenfuctions of Schrodinger operators with potentials admitting some integral conditions.T.Natsume has studied several index theorems by using the method of functional analysis.Y.Takayama and Y.Sato have studied several problems of mathematical science from the point of view of the computer science.

1.具有二次位相泛函的Feynman路径积分的Van Vleck公式在数学科学研究中占有重要地位。利用随机分析的方法，池田证明了一个与上述Van Vleck公式类似的二次位相泛函Weiner积分公式。通过用Jacpbi场描述的变换，我们可以在Wiener空间上重写与二次拉格朗日相关的二次Wiener泛函。然后，出现在Van Vleck公式中的泛函行列式可以被视为上述变换的雅可比，就像在有限维流形的情况下一样。在不使用Poincare上半平面上的热方程基本解的精确公式的情况下，N.Ikeda得到了基于Wiener空间上的热方程基本解的积分表示的Selberg迹公式的证明。T.Yamada研究了布朗局部时的主值和零能布朗可加泛函的表示。M.Arai利用泛函分析的方法研究了薛定谔算子的特征函数的增长级，得到了一些有意义的结果；T.Natsum用泛函分析的方法研究了几个指数定理；Y.Takayama和Y.Sato从计算机科学的角度研究了数学科学中的几个问题。

项目成果

M. Arai, J. Uchiyama: "Growth order of eigenfunctions of Schrodinger operators with potentials admiting some integral conditions II-Applications-" Publ. R. I. M. S. Kyoto Univ.32. 617-637 (1996)

M. Arai，J. Uchiyama：“薛定谔算子的本征函数的增长顺序，具有承认某些积分条件的潜力 II-应用-”Publ。

N. Ikeda, S.Kusuoka: "Shot time asymptotics for fundamental solutions of heat equations with boundary conditions" Proceedings of a Taniguchi International Workshop, World Scientific. 予定. (1997)

N. Ikeda、S.Kusuoka：“具有边界条件的热方程基本解的射击时间渐进”，谷口国际研讨会论文集，世界科学出版社（1997 年）。

G. A. Elliott, T. Natsume, R. Nest: "The Atiya-Singer index theoremas passage to the classical limit on quantum mechanics" Comm. Math. Phys.182. 505-533 (1996)

G. A. Elliott、T. Natsume、R. Nest：“Atiya-Singer 指数定理通向量子力学的经典极限” Comm。

G.A.Elliott,R.Nest T.Natsume: "The Atiya-Singer index theoremas passage to the classical limit on quantum mechanics" Comm.Math.Phys.182. 505-533 (1996)

G.A.Elliott、R.Nest T.Natsume：“Atiya-Singer 指数定理通向量子力学的经典极限”Comm.Math.Phys.182。

Y. Takayama: "Extraction of concurrent processes from higher dimensional automata" Lecture Notes in Computer Science, Springer. 1059. 72-86 (1996)

Y. Takayama：“从高维自动机中提取并发过程”计算机科学讲义，施普林格。