Random, Stochastic, and Self-Similar Equations

University Of Connecticut, Storrs CT

Investigators

Abstract

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.