Random, Stochastic, and Self-Similar Equations

该项目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，随机障碍物通道中的信号传播，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，该项目将为这些过程创造新的见解。该项目将教育和研究与本科生结合起来，特别是REU暑期本科生研究项目，并将基础数学研究纳入大学常规课程。该项目的更广泛影响包括以下方面。该项目将通过向本科生和研究生讲授数学研究的前沿，培养他们对现实世界与复杂数学抽象之间的新联系的理解，从而促进科学和工程人力资源的发展，这些联系是基于专门为此目的量身定制的具体问题，也是更广泛的研究计划的自然组成部分。该项目将包括采取具体步骤，扩大代表性不足群体对数学研究的参与。所有这些都将加强研究和教育的整体基础设施。自相似随机游走、扩散和狄利克雷形式的存在性和唯一性将被建立在数学的各个领域，如自相似群的极限集和Julia集的空间的广泛类别。对这些空间将得到高斯和非高斯热核估计和格林函数估计。此外，将开发工具来研究粗糙空间的可测量黎曼几何，没有曲率边界，而是使用对称扩散，狄利克雷形式和非交换概率。这些想法将应用于当前的数学物理问题，例如分形上的磁性和量子波，其长期目标是理解分形和更一般的非光滑设置中物理上有意义的路径积分。

The project contributes to the study of processes in disordered media (fractals), which have many applications in physics, chemistry, the 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 are just a few of many examples of such processes, for which the project will be creating new insights. The project integrates education and research with undergraduate students, in particular, REU summer undergraduate research projects, and incorporation of basic mathematical research in regular university courses. The broader impacts of the project include the following. The project will contribute to the development of human resources in science and engineering by teaching undergraduate and graduate students about frontiers of mathematical research, developing their understanding of new connections between real world and complicated mathematical abstractions based on concrete questions that are specifically tailored for this purpose, and also are natural parts of a broader research program. The project will include concrete steps to expand participation of underrepresented groups in mathematical research. All this will enhance overall infrastructure for research and education.The existence and uniqueness of self-similar random walks, diffusions, and Dirichlet forms will be established for a wide class of spaces arising in various areas of mathematics, such as the limit sets of self-similar groups and Julia sets. Gaussian and non-Gaussian heat kernel estimates and Green's-function estimates will be obtained for such spaces. Furthermore, tools will be developed for the study of measurable Riemannian geometry of rough spaces without curvature bounds, but instead using symmetric diffusions, Dirichlet forms, and non-commutative probability. These ideas will be applied to the current problems of mathematical physics, such as magnetism and quantum waves on fractals, with the long term goal to understand the physically meaningful path integrals in fractal and more general non-smooth settings.