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Stochastic analysis of sub harmonic functions and its application to value distribution theory

次谐波函数的随机分析及其在价值分布理论中的应用

基本信息

批准号：
10640167
负责人：
ATSUJI Atsushi
金额：
$ 1.98万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

项目摘要

It was a staring point of this project that Takegoshi obtained a criterion non-existence of minimal immersion inside cones using volume-growth of manifolds. It involves a criterion on validity of Omori-Yau maximum principle. The work led Atsuji to probabilistic research on this subject. He showed that stochastic completeness of manifolds implies this property and showed the general result of this problem which includes all of the earlier works. He also considered global behavior of minimal submanifolds by tools from stochastic analysis. It enables us to know some relationships between global behavior of Brownian motion and function theoretic properties of minimal submanifolds. They also considered non-existence theorems of harmonic maps of finite energy. Takegoshi generalized Schoen-Yau's result using L^P-analysis and maximum principle. Atsuji extended Cheng-Tam-Wan's result using probabilistic methods. It involves some Liouville type theorems for subharmonic functions and a new proof of classical results using some probabilistic technique (for example, ratio ergodic theorem). The other results of this project are obtained as follows. Komatsu studied boundary singularity of Bergman kernel and Szego kernel on strictly convex domains with smooth boundary. He determined CR invariants of weight 5 in two dimensional case. Using this result he also obtained the best result on the asymptotic expansion of these kernels. Kaneko showed a Green formula in case of local Dirichlet spaces and gave a new criterion on recurrence of diffusions. He also considered stochastic analysis on p-adic field. He showed similar results of stochastic analysis to the case of Euclidean spaces. Kotani showed a limit theorem of certain signed additive functionals of Brownian motion on R.It is a generalization of Sinai's result. He obtained some results of KdV equations with random initial data from a viewpoint of infinite soliton.

该项目的一个出发点是 Takegoshi 使用流形的体积增长获得了锥体内部最小浸入不存在的标准。它涉及大森丘极大值原理的有效性标准。这项工作促使 Atsuji 对这个主题进行了概率研究。他证明了流形的随机完备性暗示了这一性质，并展示了这个问题的一般结果，其中包括所有早期的工作。他还通过随机分析工具考虑了最小子流形的全局行为。它使我们能够了解布朗运动的全局行为与最小子流形的函数理论性质之间的一些关系。他们还考虑了有限能量调和图的不存在定理。 Takegoshi 使用 L^P 分析和极大值原理推广了 Schoen-Yau 的结果。 Atsuji 使用概率方法扩展了 Cheng-Tam-Wan 的结果。它涉及一些次调和函数的刘维尔型定理以及使用某种概率技术（例如，比率遍历定理）的经典结果的新证明。本项目的其他成果如下。 Komatsu 研究了具有光滑边界的严格凸域上的 Bergman 核和 Szego 核的边界奇异性。他确定了二维情况下权重 5 的 CR 不变量。利用这个结果，他还获得了这些核的渐近展开的最佳结果。 Kaneko 展示了局部狄利克雷空间情况下的格林公式，并给出了扩散重现的新标准。他还考虑了 p-adic 场的随机分析。他展示了与欧几里得空间情况类似的随机分析结果。 Kotani证明了R上布朗运动的某些带符号加性泛函的极限定理，它是Sinai结果的推广。他从无限孤子的角度得到了带有随机初始数据的KdV方程的一些结果。