Project/Area Number  10640167 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Basic analysis

Research Institution  Osaka University 
Principal Investigator 
ATSUJI Atsushi Graduate School of Science, Osaka Univ. Lecture, 大学院・理学研究科, 講師 (00221044)

CoInvestigator(Kenkyūbuntansha) 
KANEKO Hiroshi Graduate School of Science, Osaka Univ. Research Associate, 理学部, 助教授 (90194919)
SATAKE Ikuo Graduate School of Science, Osaka Univ. Research Associate, 大学院・理学研究科, 助手 (80243161)
TAKEGOSHI Kensho Graduate School of Science, Osaka Univ. Associate Professor, 大学院・理学研究科, 助教授 (20188171)
KOMASTSU Gen Graduate School of Science, Osaka Univ. Associate Professor, 大学院・理学研究科, 助教授 (60108446)
KOTONI Shinichi Graduate School of Science, Osaka Univ. Professor, 大学院・理学研究科, 教授 (10025463)

Project Fiscal Year 
1998 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥3,100,000 (Direct Cost : ¥3,100,000)
Fiscal Year 2000 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1999 : ¥900,000 (Direct Cost : ¥900,000)
Fiscal Year 1998 : ¥1,400,000 (Direct Cost : ¥1,400,000)

Keywords  subharmonic function / Brownian motion / harmonic map / minimal surface / Nevanlinna theory / diffusion process / martingale / Nevarlinna theory / 劣調和関数 / 極小曲面 / ブラウン運動 / 調和写像 / 値分布論 / Brown運動 / 最大値原理 / P調和写像 / 準線形楕円形作用素 / ディリクレ空間 
Research Abstract 
It was a staring point of this project that Takegoshi obtained a criterion nonexistence of minimal immersion inside cones using volumegrowth of manifolds. It involves a criterion on validity of OmoriYau 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 nonexistence theorems of harmonic maps of finite energy. Takegoshi generalized SchoenYau's result using L^Panalysis and maximum principle. Atsuji extended ChengTamWan'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 padic 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.
