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.

一系列广泛的数学方法将被用来增加在自相似，分形和无序介质中发生的过程的长期和短期行为的理解。自相似Dirichlet形式，拉普拉斯算子和扩散的存在性和唯一性将在广泛的一类分形上得到证明，包括无限分歧的广义Sierpinski地毯和自相似群的极限集。高斯和非高斯热核估计和绿色的功能估计将研究自相似和随机分形。该项目将有助于遍历理论的产品不一定独立的矩阵和它们的关系，局部性质的分形过程。本文将给出具有小扰动的微分方程和差分方程的李雅普诺夫指数的渐近公式，以及随机微分方程的李雅普诺夫指数的估计，并将其与随机微分方程的谱问题联系起来。工作将做调查等问题的功能空间，偏微分方程，以及各种概念的微分几何和拓扑分形。该项目有助于更好地理解Julia集、自相似群和有限自动机的极限集、量子图、矩阵和遍历理论的乘积、非交换微积分和几何的分析，有助于研究无序介质（分形）中的过程，这些过程在物理、化学、生物科学和工程中有许多应用。渗透簇中的扩散过程、分形物体的振动、信号在具有随机障碍物的通道中传播、分形天线中的电磁波、海洋学中的罗斯比波、金融市场模型只是此类过程的许多例子中的几个。该项目包括将研究和教育结合起来的各种活动。该项目更广泛的影响包括促进科学和工程领域的人力资源开发，扩大代表性不足群体的参与，以及加强研究和教育基础设施。