Random, Stochastic, and Self-similar Equations

将使用广泛的数学方法来增加对发生在自相似、分形和无序介质中的过程的长期和短期行为的理解。在包括无限分枝的广义Sierpinski地毯和自相似群的极限集在内的一类广泛的分形上，证明了自相似Dirichlet形式、Laplace算子和扩散的存在唯一性。高斯型和非高斯型热核估计以及格林函数估计将基于自相似和随机分形学进行研究。该项目将有助于遍历理论，即不一定独立的矩阵的乘积及其与分形过程的局部性质的关系。给出了具有小随机扰动的微分方程组和差分方程组的Lyapunov指数的渐近公式，以及随机微分方程谱问题的Lyapunov指数估计。将致力于研究诸如泛函空间、偏微分方程以及关于分数形的各种微分几何和拓扑学概念。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。该项目有助于研究无序介质(分形学)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。渗流簇中的扩散过程、分形物体的振动、带有随机障碍物的通道中的信号传播、分形天线中的电磁波、海洋学中的Rossby波、金融市场模型等都是此类过程的众多例子中的一小部分。该项目包括各种将研究和教育相结合的活动。该项目的更广泛影响包括对科学和工程人力资源开发的贡献，扩大代表性不足群体的参与，以及加强研究和教育的基础设施。

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.