Research on Probability and Related Topics of Analysis

概率及相关分析主题研究

• 批准号: 07454034
• 负责人: TANAKA Hiroshi
• 金额: $ 2.88万
• 依托单位: Keio University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1995
• 资助国家: 日本
• 起止时间: 1995 至 1996
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07454034/
• 关键词: Brownian Motion; Diffusion Process; Operator-Self-Similar Process; Limit Theorem; Rigidity of Group Actions; Transcendental Number; Ergodic Transformation; Large Deviation; ホワイトノイズ媒質; 流体力学極限; 自己相似過程; ナビェ・ストークス方程式; 自由境界問題; ランダム媒質

1. (1) New results for long time asymptotic behavior of a Brownian motion (diffusion process) with a constant positive drift coefficient kappa in a white noise environment Were obtained by H.Tanaka ; the results vary with kappa. (2) New results were obtained (i) in the limit theorem of operator-self-similar processes (by M.Maejima), (ii) in a problem of large deviation of Nonsymmetric Markov processes and (iii) for the construction and hydrodynamic limit of a certain spin system on Z (by Y.Suzuki).2. Y.Ito clarified some number-theoretic structure of "infinite sets of integers" arising as an invariant in the classification of ergodic transformations without finite invariant measures, in particular, of the types II_* and III.3. A new approach based on stochastic analysis was poposed by M.Kanai in the investigation of group actions. Y.Maeda investigated certain groups of diffeomorphisms introducing the associated zeta functions.4. Some new results concerning local and global solutions of the incompressible Navier-Stokes equations (by A.Tani).5. I.Shiokawa (and others) investigated the transcendentality problem of certain numbers connected with Jacobi's theta series or with Fibonacci sequences, solving a conjecture by Liouville (1851) on transcendental numbers.

1. (1)H.Tanaka对白色噪声环境中具有常数正漂移系数kappa的布朗运动（扩散过程）的长时间渐近行为给出了新的结果，其结果随kappa的变化而变化. (2)在算子自相似过程的极限定理（M.Maejima），非对称马氏过程的大偏差问题，Z上某自旋系统的构造和流体动力学极限（Y.Suzuki）等方面获得了新的结果.伊藤澄清了一些数论结构的“无限集的整数”出现作为一个不变的分类遍历变换没有有限不变的措施，特别是，类型II_* 和III.3。金井先生在研究群体行为时提出了一种基于随机分析的新方法。Y.Maeda研究了引入相关zeta函数的某些群的自同态。关于不可压Navier-Stokes方程局部解和整体解的一些新结果（A.Tani）. I.Shiokawa（和其他人）研究了某些与Jacobi的θ级数或Fibonacci序列有关的数的超越性问题，解决了Liouville（1851）关于超越数的猜想。

项目成果

E.Bolthausen: "Laplace approximations for large deviations of nonreversible Markov processes I : The non-degenerate case" Ann.Probab.23. 235-267 (1995)

E.Bolthausen：“不可逆马尔可夫过程大偏差的拉普拉斯近似：非简并情况”Ann.Probab.23。

Yuji Ito: "Direct sum decomposition of the integers" Tokyo J. Math.18. 259-270 (1997)

Yuji Ito：“整数的直接和分解”Tokyo J. Math.18。

H.Tanaka: "Environment-wise central limit theorem for a diffusion in a Brownian environmnet with large drift" Ito's Stochastic Calculus and Probab.Th.N.Ikeda et al.eds.,Springer Verlag. 373-384 (1996)

H.Tanaka：“大漂移布朗环境中扩散的环境中心极限定理”Ito 的随机微积分和 Probab.Th.N.Ikeda 等人，Springer Verlag。

E. Bolthausen, J. Deuschel and Y. Tamura: "Laplace approximations for large deviations of nonreversible Markov processes Part I: The nondegenerate case" Annals of Probability. 23. 236-267 (1995)

E. Bolthausen、J. Deuschel 和 Y. Tamura：“不可逆马尔可夫过程大偏差的拉普拉斯近似，第一部分：非简并案例”《概率年鉴》。

N,Tanaka,A,Tani: "Large-time existence of surface wave in incompressible viscous fluids with or without surface tension" Arch.Rat.Mech.Anal.130. 303-314 (1995)

N，Tanaka，A，Tani：“在有或没有表面张力的不可压缩粘性流体中表面波的大量存在”Arch.Rat.Mech.Anal.130。