Limit Theorems for Non-Stationary and Random Dynamical Systems

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Although much is understood about the statistical properties of physical systems in equilibrium, many systems of interest cannot be modeled realistically as being in equilibrium, meaning that the chance of observing an event or taking a certain measurement on the system does not change over time. Some physical systems are subject to external influences that vary over time; for example, the atmosphere is affected by the seasonal ocean temperatures, and the state of an electrical grid is affected by the time-varying demand for power. Physical systems are often also subject to random effects, either through measurement reading errors or noise in the system itself. This research project aims to develop a good statistical understanding of non-equilibrium and random complex systems using recent advances in probability and analysis. A better understanding of the statistical properties of non-equilibrium systems and random systems will lead to improved predictions and expand our knowledge of the statistical laws that can be expected to hold in nature. The project aims to develop new techniques based on recent developments in martingale theory and analysis to investigate central limit theorems, almost sure invariance principles (a strong form of approximation by Brownian motion), large deviations, and other advanced statistics for a wide class of non-equilibrium and random dynamical systems that are used to model complex physical systems. Examples of the systems to be studied include billiard systems (polynomially and exponentially mixing), models of intermittency, Lorenz-like maps and flows, and systems arising from statistical mechanics. Statistical properties of random dynamics can be investigated by sampling over all noise realizations (annealed statistics) or along individual realizations of the noise (quenched statistics). The project will also develop our understanding of annealed and quenched statistical properties and the interplay between the two.