MATEMATICAL ANALYSIS OF INFINITE DIMENSIONAL STOCHASTIC 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。

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.