Study on global properties of local martingales and martingales on manifolds.

局部鞅和流形上鞅的全局性质研究。

• 批准号: 13640170
• 负责人: ATSUJI Atsushi
• 金额: $ 2.24万
• 依托单位: Keio University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640170/
• 关键词: Brownian motion; local martingale; T-Martingale; δ-subharmonic function; Liouville theorem; missional surface; Nevanlinna theory; subharmonic functions; Brownian motion; δ-sabharmonic function; Lienville theorem; マルチンゲール; ブラウン運動; 劣調和関数; ネヴァン

Head investigator A.Atsuji obtained the following main three results.1. For a minimal surface which may not be properly immersed, its total curvature is finite if its projective volume is finite. In this result we used some properties of Brownian motion on the surfaces which is martingale on the ambient spaces. Lemma for logarithmic derivative of δ-subharmonic functions which is proved by using properties of 1-dimensional local martingale works well.2. Parabolicity is characterized by δ-subharmonic functions. Using this characterization we showed some Liouville type theorems for harmonic maps of finite energy from parabolic manifolds to Hadamard manifolds. We used some properties of martingales given as images of Brownian motion by harmonic maps.3. A Nevanlinna theory for meromorphic functions on complex submanifolds in C^n is obtained. S.Kotani gave a necessarily and sufficient condition for 1 dimensional diffusions to be true martingales. K.Takegoshi gave some criteria for complete Riemannian manifolds to be parabolic, Liouville type theorems for harmonic maps and some results on isometry of conformal metric of positive scalar curvature. Y.Suzuki same results on long time behavior of diffusion processes in random environment with one-sided Brownian potential.

头部研究者A.ATSUJI获得了以下三个结果。1 Martingale的工作良好。 C^获得了S.Kotani的必要条件，并使1个Dimesional扩散为寄生虫.suzuki在具有单侧布朗电势的随机环境中的扩散过程的长时间行为相同。

项目成果

Atsushi Atsuji: "A lemma of logarimic derivative for some S-subharmonic functions"Complex variables. 46. 195-206 (2001)

Atsushi Atsuji：“某些 S 分谐波函数的对数导数引理”复变量。

Kensho Takegoshi: "A note on divergence of Lp-integrals of subharmonic functions and its applications."Proc.Amer.Math.Soc.. vol.131,no.9. 2849-2858 (2003)

Kensho Takegoshi：“关于分调和函数的 Lp 积分的散度及其应用的说明。”Proc.Amer.Math.Soc.. vol.131，no.9。

Atsushi Atsuji: "A lemna of logatrithmic derivative for some f-subharmonic functions"Complex Variables. 46. 195-206 (2001)

Atsushi Atsuji：“某些 f 次谐波函数的对数导数的浮理”复变量。