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

takgoshi利用流形的体积生长获得了锥内最小浸没不存在的准则，这是这个项目的起点。它涉及到一个关于Omori-Yau极大原则有效性的判据。这项工作使Atsuji开始对这一主题进行概率研究。他证明了流形的随机完备性隐含了这一性质，并给出了这个问题的一般结果，其中包括了所有早期的工作。他还用随机分析的方法研究了最小子流形的全局行为。它使我们认识到布朗运动的全局行为与极小子流形的泛函理论性质之间的一些关系。他们还考虑了有限能量调和映射的不存在定理。Takegoshi用L^ p分析和极大值原理推广了Schoen-Yau的结果。Atsuji用概率方法扩展了Cheng-Tam-Wan的结果。它涉及到一些次调和函数的Liouville型定理，以及使用一些概率技术（如比率遍历定理）对经典结果的一个新的证明。本项目的其他结果如下：Komatsu研究了边界光滑的严格凸域上Bergman核和Szego核的边界奇异性。他确定了二维情况下权值为5的CR不变量。利用这一结果，他也得到了这些核的渐近展开式的最佳结果。Kaneko给出了局部Dirichlet空间的Green公式，并给出了扩散递推的新判据。他还考虑了p进场的随机分析。他展示了与欧几里得空间类似的随机分析结果。Kotani给出了r上布朗运动的某些带符号加性泛函的极限定理，这是对Sinai结果的推广。从无限孤子的观点出发，得到了具有随机初始数据的KdV方程的一些结果。

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.