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.
The final report for the project includes 30 completed papers (21 published or accepted and 9 preprints). The results mostly followed the path outlined in the summary of the project. In particular, an wide array of mathematical methods have been 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 have been 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 have been studied on self-similar and random fractals. The project has contributed 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 have been investigated, and related to the spectral problems for stochastic differential equations. Work has been done to investigate such questions as functional spaces, partial differential equations, and various notions of differential geometry and topology on fractals. The project has contributed to better understanding of 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. In terms of the broader impacts, the project contributed 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 included various activities that integrate research and education. Training and Development included training of graduate students in research and integrating research and education, work with undergraduate REU students, and work with undergraduate students within innovative Mathematics Scholar course (several research papers with undergraduate students have been published).The broader impacts of the project included contribution to the development of human resources in science and engineering, expanding participation of underrepresented groups, and enhancing infrastructure for research and education.
