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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.

武越利用流形的体积增长得到了锥内最小浸入不存在的一个判据，这是本课题的出发点。它涉及Omori-Yau极大值原理有效性的一个判据。这项工作导致笃二概率研究这个问题。他表明，随机完备性的流形意味着这一性质，并显示了一般结果，这个问题，其中包括所有的早期工作。他还认为全球行为的最小子流形的工具，从随机分析。它使我们了解布朗运动的整体性态与极小子流形的函数论性质之间的某些关系。他们还考虑了有限能量调和映射的不存在定理。武越利用L^P分析和极大值原理推广了Schoen-Yau的结果。Atsuji用概率方法推广了Cheng-Tam-Wan的结果。它涉及到次调和函数的Liouville型定理和经典结果的概率证明（如比率遍历定理）。本项目的其他成果如下。Komatsu研究了边界光滑的严格凸区域上Bergman核和Szego核的边界奇异性。他确定CR不变量的重量5在二维情况下。利用这一结果，他还获得了最好的结果渐近扩展这些内核。Kaneko在局部Dirichlet空间的情形下给出了一个绿色公式，并给出了扩散常返性的一个新判据.他还认为随机分析的p进领域。他表现出类似的结果随机分析的情况下，欧几里德空间。Kotani证明了R上布朗运动的某些符号可加泛函的一个极限定理，它是Sinai结果的推广。他从无限孤子的观点出发，得到了具有随机初值的KdV方程的一些结果。