AMC-SS: Statistical Analysis and Modeling of Complex Physical Systems

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The project will investigate key statistical properties such as large deviation estimates, extreme value statistics and the local central limit theorem for dynamical systems with some degree of hyperbolicity. The dynamics and statistical properties of coupled non-uniformly hyperbolic maps will be studied. Non-stationary random walk models, and related Fokker-Planck equations, will be developed to describe time-inhomogeneous systems such as currency-exchange markets and highly turbulent flows. In addition, characterizations of spatio-temporal patterns will be exploited to study epitaxial and domain growth in magnetic materials.Complex systems - examples range from weather systems and turbulent fluids to financial markets and animal populations - are often best understood in terms of their statistical properties, that is, by trying to provide estimates of the probability of certain types of behavior. A good statistical understanding of many of these systems is lacking because the usual modeling assumptions, for example independence, are not satisfied. The proposal aims to deepen our knowledge of the statistics of a broad range of complex systems, to enhance our ability to predict their behavior and assess the risk of certain events. In addition the proposal will develop mathematical models to describe financial markets and turbulence as well as provide improved statistical characterizations of disordered patterns in magnetic materials.