Random Dynamical Systems and Limit Theorems for Optimal Tracking

Dynamical systems serve as important mathematical models for a wide variety of physical phenomena, arising in such areas as weather modeling, systems biology, and statistical physics. A dynamical system consists of a state space, in which a point represents a complete description of the state of the system, and a rule governing the evolution of the system from one state to another. This project focuses on the long-term behavior of such systems from two complementary points of view. From the first point of view, the project seeks to describe the behavior of typical systems when the rules of evolution are chosen at random. Such results shed light on what properties one might expect to find in disordered systems. The second point of view, the 'inverse problem,' concerns the statistical problem of recovering some information from the observation of a dynamical system. While there are many examples of dynamical systems being used as mathematical models, and there is a large statistical literature regarding inference and estimation, the performance of statistical procedures when applied to data generated by nonlinear dynamical systems is poorly understood. This project focuses on characterizing when traditional statistical procedures may be effectively applied in the context of dynamical systems. Beyond the very fertile potential applications, the project will also have broader impact on training of graduate students who will acquire invaluable skills in sound probabilistic modeling and statistical inference by working on the project's research topics.

Symbolic dynamical systems, which may be defined in terms of discrete constraints on the possible trajectories, serve as prototypical models of systems that evolve over time. Random ensembles of these systems may be produced by selecting the constraints at random. Ideas from both discrete probability and dynamical systems may then be used to analyze the structural properties of the resulting systems with high probability. For the inverse problem, this project seeks to evaluate the performance of several statistical inference procedures, both frequentist and Bayesian, when the models involved are dynamical systems. Fundamental questions about convergence and consistency of the procedures may be addressed using tools from ergodic theory, such as joinings and the thermodynamic formalism.