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

首席研究员A.Atsuji获得了以下三个主要结果。对于一个可能没有适当浸入的极小曲面，如果它的射影体积是有限的，那么它的全曲率是有限的。在这一结果中，我们利用了曲面上的布朗运动的一些性质，它是环境空间上的鞅。利用一维局部鞅的性质证明了δ-亚调和函数的对数导数引理。抛物性用δ-亚调和函数来刻画。利用这个刻划，我们证明了从抛物流形到Hadamard流形的有限能量调和映射的一些Liouvile型定理。利用由调和映射给出的布朗运动像所给出的一些性质。给出了C^n中复子流形上的亚纯函数的一个Nevanlinna理论。S.Kotani给出了一维扩散为真鞅的充要条件。K.Takegoshi给出了完备黎曼流形是抛物型的一些判据，调和映射的Liouvile型定理，以及关于正标量曲率的共形度量等距的一些结果。Y.Suzuki同样研究了具有单边Brown势的随机环境中扩散过程的长时间行为。

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.