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.
