MATEMATICAL ANALYSIS OF INFINITE DIMENSIONAL STOCHASTIC MODELS

无维随机模型的数学分析

• 批准号: 13640103
• 负责人: SHIGA Tokuzo
• 金额: $ 1.92万
• 依托单位: Tokyo Institute of Technology
• 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-13640103/
• 关键词: Strong Disorder; Anderson Localization; Fermion Random Field; Critical Phenomena; アンダーソン局在; フェルミオン確率場; 臨界現象; リヤプノフ指数; 向きをもつポリマーモデル; Strong disorder; フェルミオンランダム場; 条件付極限定理; Bessel meander

We performed this research project on "Mathematical Analysis of Infinite Dimensional Stochastic Models", and obtained the following results.1.Asymptotics of Lyapunov exponent of Anderson model : It is established that the discrete heat equation with space-time white noise potential over the discrete Euclidean space, so salled "Parabolic Anderson model" has a Lyapunov exponent exhibiting the exponential order of the solution, which is independent of initial conditions. Morever we developed asymptotic analysis of the exponent with respect to the coupling constant and obtained precise asymptotic formula.2.Directed Polymers in Random environment : The theory of directed polymers in random environments is still underdeveloping, particularly in low dimensional cases. We established equivalence of local property of the sample path and some decay order of the random partition functions.3.One of co-workers of this project investigated spectrum structure of random Schroedinger operator over line graph and proved "Anderson localization", another one investigated ergodic properties of the shift dynamics associated with fermion random fields. The other one contributed to the rigorous theory of critical phenomena for stochastic statistical mechanics in establishing presice asymptotic order of two point functions in self-avoiding random walk models and related models.

本论文的主要研究内容是“无穷维随机模型的数学分析”，主要研究结果如下：1.安德森模型的李雅普诺夫指数的渐近性：在离散欧氏空间上建立了具有时空白色噪声势的离散热方程，即所谓的抛物型安德森模型，其解的李雅普诺夫指数为指数级，这与初始条件无关。其次，我们对指数关于耦合常数的渐近分析，得到了精确的渐近公式。2.随机环境中的定向聚合物：随机环境中的定向聚合物的理论还不成熟，特别是在低维情况下。3.本项目的一个合作者研究了线图上随机Schroedinger算子的谱结构并证明了“安德森定域性”，另一个合作者研究了费米子随机场移动动力学的遍历性。另一个贡献是建立了自避免随机游动模型及相关模型中两点函数的preice渐近阶，从而为随机统计力学的严格临界现象理论做出了贡献。

项目成果

S.N.Ethier, T.Shiga: "Fleming-Viot process with unbounded selection, II"Proceeding on Markov Processes and Controlled Markov Chains Kluwer Acad. Pub., eds Z.Hou, J.A.Filar and A.Chen. 123. 234-240 (2002)

S.N.Ethier、T.Shiga：“无界选择的 Fleming-Viot 过程，II”马尔可夫过程和受控马尔可夫链 Kluwer Acad 进展。

Z.Li, T.Shiga, M.Tomisaki: "A Conditional Limit Theorem for Generalized Diffusion Processes"Journal of Mathematics of Kyoto University. (掲載予定). (2003)

Z.Li、T.Shiga、M.Tomisaki：“广义扩散过程的条件极限定理”京都大学数学杂志（即将出版）。

F.Nakano, Y.Nomura: "Random magnetic fields on line graphs"Journal of Mathematical Physics. 44. 4988-5002 (2003)

F.Nakano，Y.Nomura：“线图上的随机磁场”数学物理杂志。

T.Hara, R.van der Hofstad, G.Slade: "Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models."Annals of Probability. 31. 349-408 (2003)

T.Hara、R.van der Hofstad、G.Slade：“临界两点函数和展开高维渗滤及相关模型的花边展开。”概率年鉴。

Yu.Higuchi, T.Shirai: "Weak Bloch property for the discrete magnetic Schr\"{o}dinger operators"Nagoya Math.J.. 161. 127-154 (2001)

Yu.Higuchi, T.Shirai：“离散磁 Schr 的弱 Bloch 性质”{o}dinger 算子”Nagoya Math.J.. 161. 127-154 (2001)