Random, Stochastic, and Self-similar Equations

随机、随机和自相似方程

• 批准号: 0806103
• 负责人: Alexander Teplyaev
• 金额: $ 17万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0806103&HistoricalAwards=false
• 关键词: Random; Stochastic; Self; similar; Equations

A wide array of mathematical methods will be used to increase the understanding of the long and short term behavior of processes occurring in self-similar, fractal and disordered media. The existence and uniqueness of self-similar Dirichlet forms, Laplacians and diffusions will be proved on a wide class of fractals, including infinitely ramified generalized Sierpinski carpets and limit sets of self-similar groups. Gaussian and non-Gaussian heat kernel estimates and Green's function estimates will be studied on self-similar and random fractals. The project will contribute to the ergodic theory of products of not necessarily independent matrices and their relation to local properties of processes on fractals. Asymptotic formulas for Lyapunov exponents of differential and difference equations with small random perturbations, and estimates of the Lyapunov exponents of stochastic differential equations will be obtained, and related to the spectral problems for stochastic differential equations. Work will be done to investigate such questions as functional spaces, partial differential equations, and various notions of differential geometry and topology on fractals. The project contributes to better understand the analysis on Julia sets, limit sets of self-similar groups and finite automata, quantum graphs, products of matrices and ergodic theory, non-commutative calculus and geometry.The project contributes to the study of processes in disordered media (fractals), which have many applications in physics, chemistry, biological sciences and engineering. Diffusion processes in percolation clusters, vibrations of fractal objects, signal propagating in channels with random obstacles, electro-magnetic waves in fractal antennae, Rossby waves in oceanography, models of financial markets are just a few of many examples of such processes. The project includes various activities that integrate research and education. The broader impacts of the project include contribution to the development of human resources in science and engineering, expanding participation of underrepresented groups, and enhancing infrastructure for research and education.

多种数学方法将用于增加对自相似，分形和无序培养基中发生的过程的长期和短期行为的理解。自相似的迪里奇，拉普拉斯主义者和扩散的存在和独特性将在一系列宽类的分形上证明，包括无限分支的广义sierpinski地毯和限制自相似群体的限制集。高斯和非高斯热核估计值和格林的功能估计将在自相似和随机分形上进行研究。该项目将有助于不一定是独立矩阵的产品及其与分形过程的局部特性的关系。将获得与随机扰动小的差异方程式和差异方程式的Lyapunov指数的渐近公式，并将获得随机微分方程的Lyapunov指数的估计，并与随机微分方程的光谱问题有关。将进行研究，以调查功能空间，部分微分方程以及分形差异几何学和拓扑概念等问题。该项目有助于更好地了解对朱利娅集合的分析，自相似群体的限制集以及有限的自动机，量子图，矩阵和千古论理论的产物，非共同的微积分和几何形状。该项目有助于无序培养基（Fractals）过程（Fractals）的过程，这些过程在物理，化学，化学，生物学，生物学和工程中都有许多应用。渗透簇，分形物体的振动，信号在具有随机障碍物的通道中传播的传播，分形触角中的电磁波，海洋学中的rossby波，金融市场模型只是此类过程的许多例子中的一些例子。该项目包括整合研究和教育的各种活动。该项目的更广泛影响包括对科学和工程学中人力资源的发展，扩大代表性不足的群体的参与以及增强研究和教育的基础设施的贡献。