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

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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.