Random, Stochastic, and Self-similar Equations

随机、随机和自相似方程

• 批准号: 1106982
• 负责人: Alexander Teplyaev
• 金额: $ 32.09万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-15 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1106982&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 random processes occurring in self-similar, fractal and disordered media. The existence and uniqueness of self-similar Dirichlet forms, diffusions, and random walks will be proved on a wide class of fractals, including infinitely ramified fractals appearing as limit spaces of groups. Gaussian and non-Gaussian heat kernel estimates and Green's function estimates will be studied on disordered systems, such as self-similar and random fractals. Furthermore, probabilistic tools will be developed to study non-commutative analysis on and generalized differential geometry of disordered spaces that carry a local Dirichlet form. In addition, 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 and wave propagation in fractal and other disordered media.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 propagation in channels with random obstacles, electro-magnetic waves in fractal antennae, Rossby waves in oceanography, models of financial markets, neural structures are just a few of many examples of such processes. Thus the project contributes to the integration of mathematics, physics, biological sciences and engineering. The project integrates education and research with undergraduate students. The broader impacts of the project include contributions 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随机过程的长期和短期行为的理解。自相似的差异形式，扩散和随机步行的存在和唯一性将在一类宽类的分形上证明，包括无限分支的分形，这些分形出现为群体的极限空间。高斯和非高斯热核估计值和格林的功能估计将在无序系统（例如自相似和随机分形）上进行研究。此外，将开发概率工具来研究携带局部迪利奇形式的无序空间的非交通分析和广义差异几何形状。此外，该项目将有助于不一定是独立矩阵的产品的千古理论及其与分形过程的局部特性的关系。将获得差异和差异方程式的Lyapunov指数的渐近公式，这些方程式较小，随机扰动较小，以及对随机微分方程的Lyapunov指数的估计，以及与随机微分方程方程和波浪传播的光谱问题有关的频谱问题，对分形和其他无序的媒体中的影响促进了许多序列的序列（序列）（序列中）（序列化的序列）（有效的序列）（有效的序列），循环（有效），有效期（有效），是有效的，有效的是，有效的是，有效的序列，是有效的。生物科学和工程。渗透簇，分形对象的振动，具有随机障碍物的通道，分形触角的电磁波，海洋学中的罗斯比波，金融市场模型，神经结构模型的传播的传播过程，只是许多示例。因此，该项目有助于数学，物理，生物科学和工程学的整合。该项目将教育和研究与本科生相结合。该项目的更广泛影响包括对科学和工程中人力资源发展的贡献，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。