Microlocal analysis and pseudo-differential operators

微局部分析和伪微分算子

• 批准号: 14340045
• 负责人: KOTANI Shinichi
• 金额: $ 7.3万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14340045/
• 关键词: dispersive equation; index theorem; smoothing property; Wavelet; Burgers equation; Schroedinger equation; KdV equation; Krein's theory; 超局所解析; 擬微分作用素; マルチンゲール; シュレーディンガー作用素; スペクトル; 超関数; 波面集合; L^p(R^n)-有界性; 表象; 磁場ポテンシャル; 量子化

1.Differential equations (Uchida, Iwasaki, Sugimoto)(1)The local index theorem was extended to manifolds with boundary and Koeler manifolds by applying heat equations.(2)Hoermander's uniqueness theorem was proved geometrically by formulating his strong pseudo-convex condition for general differential equation systems.(3)The smoothing property for dispersive equations was considered by transforming by simpler systems through associated dynamical systems.2.Numerical analysis and applications of pseudo-differential operators (Ashino)(1)Decomposition of singular values and wavelet analysis were applied to image analysis.(2)Wavelet was applied to identify the systems.3.Probability theory (Kotani, Isozaki)(1)By extending the deviation theorem, a theorem including three random variables was obtained and was applied to the investigation of the distribution of the hitting time to the half line of 2-dim. random walks.(2)A final answer for the distribution of clusters created by Burgers equations with random initial values was obtained.(3)Higher dimensional Schroedinger operators with random potentials was studied by introducing an exponents of the Green functions corresponding to the Lyapounov exponents in 1-dim.. For 1-dim. operators, the KdV-flow was constructed by employing Sato's theory.(4)1-dim. diffusion processes were studied from the point of view of martingales and Krein's spectral theory.

1.微分方程（内田，岩崎，杉本）（1）利用热方程将局部指标定理推广到有边流形和Koeler流形。（2）通过对一般微分方程组的强伪凸性条件的表述，给出了Hoermander唯一性定理的几何证明。（3）通过关联动力系统的简单系统变换，研究了色散方程的光滑性。2.伪微分算子（Ashino）的数值分析与应用（1）奇异值分解和小波分析在图像分析中的应用。3.概率论（Kotani，Isozaki）（1）通过推广偏差定理，得到了一个包含三个随机变量的定理，并将其应用于2维随机游动的半直线的命中时间分布的研究。(2)A得到了由Burgers方程产生的团簇在随机初值下的分布的最终答案。（3）通过引入一维空间中对应于Lyapounov指数的绿色函数的指数，研究了具有随机势的高维Schroedinger算子.对于一维算子，利用Sato理论构造了KdV流.（4）从鞅和Krein谱理论的角度研究了一维扩散过程。

项目成果

M.Sugimoto: "A smoothing property of Schrodinger equations along the sphere"J.Anal.Math.. 89. 15-30 (2003)

M.Sugimoto：“薛定谔方程沿球面的平滑特性”J.Anal.Math.. 89. 15-30 (2003)

杉本 充: "A weak extension theorem for inhomogeneous differential equations"Forum Math.. 13. 323-334 (2001)

杉本满：“非齐次微分方程的弱可拓定理”论坛数学.. 13. 323-334 (2001)

On a Condition that One-Dimensional Diffusion Processes are Martingales

• DOI: 10.1007/978-3-540-35513-7_12
• 发表时间: 2006
• 作者: S. Kotani

A family of symmetric stable-like processes and its global path properties

一族对称类稳定过程及其全局路径性质

• 发表时间: 2004
• 期刊: Probability and Mathematical Statistics 24
• 作者: Y.Isozaki;T.Uemura
• 通讯作者: T.Uemura

内田 素夫: "A non-existence theorem of lacunas for hyperbolic differential operators with constant coefficients"Ark. Mat.. 40. 201-205 (2002)

Motoo Uchida：“常系数双曲微分算子的空白不存在定理”Mat.. 40. 201-205 (2002)