Statistical Properties of Dynamical Systems

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The goal of this project is to understand the origin of stochasticity in deterministic systems, with particular attention to extended systems that can be used to model transport phenomena in physical and biological systems and to systems whose evolution laws change with time. The principal investigator will use tools from probability and geometry in order to develop methods to predict the statistical behavior of complicated systems. The project provides training for graduate students, and a significant portion of the research will be done in collaboration with graduate students, postdoctoral fellows, and other early-career mathematicians.The project consists of four parts:(I) The rate of convergence in the Central Limit Theorem;(II) Dynamics of systems preserving infinite measure;(III) Flexibility of statistical properties; and(IV) Transport of random media.The unifying theme of the project is the interplay between dynamics and probability. Probabilistic techniques, such as refined limit theorems and coupling, are used to the study of deterministic systems, and dynamical methods including spectral properties of transfer operators and renormalization dynamics on the space of lattices are used to handle probabilistic questions. Results obtained in parts (I) and (II) will be applied in parts (III) and (IV) of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.