• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to project page

2000 Fiscal Year Final Research Report Summary

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

Research Project

Project/Area Number 10640167
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionOsaka University

Principal Investigator

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

Co-Investigator(Kenkyū-buntansha) 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)
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)
Project Period (FY) 1998 – 2000
Keywordssubharmonic function / Brownian motion / harmonic map / minimal surface / Nevanlinna theory / diffusion process / martingale
Research Abstract

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.

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] Atsushi Atsuji: "A Casorati-Weierstrass theorem for holomorphic maps and invariant σ-fields holomrphic diffusions"Bull.Sci.math.1. 123. 371-383 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Atsushi Atsuji: "Remarks on harmonic maps into a cone form stochastically coyrhete manifolds"Proc.Japan Acad.. 75. 105-108 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Atsushi Atsuji: "A lemma of logarithmic drivative for some f-subharmonic functions"Complex Variables. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Atsushi Atsuji: "Nevanlinna theory and Stochastic calculus"Proc.2nd ISAA C Cory.. 427-432 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kensho Takegoshi: "A maximum principle for P-harmonic maps with Li furite energy"Proc.AMS. 126. 3749-3753 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Hiroshi KaueKo: "Recurrence and transienes criteria for symmetric Hunt process"Potential Analysis. 13. 185-197 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Atsushi Atsuji: "A Casorati-Weierstrass theorem for holomorphic maps and invariant σ-fields of holomorphic diffusions."Bull.Sci.math.. 123. 371-383 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Atsushi Atsuji: "Remarks on harmonic maps into a cone from stochastically complete manifolds."Proc. Japan Acad.. 75. 105-108 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Atsushi Atsuji: "A lemma of logarithmic derivative for some δ-subharmonic functions."Complex Variables.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Atsushi Atsuji: "Nevanlinna theory and Stochastic calculus - Someapplications of stochastic first main theorem -."Proc. 2nd ISAAC cong., H.G.W.Begehr et al.eds, Kluwer. 427-432 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kensho Takegoshi: "A maximum principle for P-harmonic maps with L^q finite energy."Proceedings of A.M.S.. 126. 3749-3753 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hiroshi Kaneko: "Recurrence and transience cirteria for symmetric Hunt processes."Potential Analysis. Vol.13, No.2. 185-197 (2000)

    • Description
      「研究成果報告書概要(欧文)」より

URL: 

Published: 2002-03-26  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi