Large Deviations in Large Non-equilibrium Systems

点击翻译按钮获取中文摘要

Central to research in modern probability theory is the study of how complex random systems behave. Besides being of fundamental interest, studying the behavior of such systems improves understanding of how heat flows through a medium, how tumors grow, how water waves propagate, and how traffic jams form and dissolve. This project focuses on a particular aspect of the behavior known as large deviations, which is about small-probability events occurring in the systems of interest. The theory of large deviations is one of the pillars of probability theory; understanding how it manifests itself in different types of systems is a practical and important task. Further, the study of large deviations has applications to understanding excursions between meta-stable states, analyzing phase transitions, and benchmarking numerical algorithms. The research aims to advance understanding in these and other areas. The project also provides research training opportunities for graduate students.In concrete terms, this project will study the finite-dimensional and sample-path large deviations of asymmetric interacting particle systems and of nonlinear stochastic partial differential equations. They exhibit large deviations in various regimes, including the short-time, intermediately-long-time, and long-time regimes. The main goals of this project are: i) establishing the short-time large deviation principles of various systems; ii) studying how the large deviations in the short-time regime converge to the (mostly conjectural) large deviations in the long-time regime; and iii) investigating the large deviations in the intermediately-long-time regime. These goals involve the mathematical study of various physical phenomena, including the correspondences between soliton waves and shock waves, the spontaneous generation of soliton waves, and dynamical symmetry breaking. The goals of this project will be achieved by a combination of stochastic analysis and integrable structures using the Feynman-Kac formula, the connection to quantum-many-body systems, and the inverse scattering transform of integrable partial differential equations.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.