Multiscale Effects and Tail Events for Infinite-Dimensional Processes and Interacting Particle Systems

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Probabilistic models are commonly used to represent physical, biological, and financial phenomena that are often too complex to solve, approximate or even simulate on computers. One of the challenges facing applied mathematics and probability is to obtain accurate and provably efficient methods to approximate and simulate a range of complex systems using probabilistic models. The primary purpose of this research is to rigorously investigate problems related to multiscale systems, rare events, and related simulation methods. The principal investigator (PI) is interested in studying stochastic dynamical systems that may have different time scales and quantifying related events that may be rare on a given time scale but can have important consequences for the system itself. The questions of interest in this project are motivated both by fundamental mathematical questions and by a broad array of questions in other branches of science. Examples range from estimation of rare event probabilities in chemical physics, hydrodynamics, coupled chemical reactions with spatially-dependent diffusion, to population genetics and opinion dynamics. This research project is integrated with an educational program that is designed to help in the training of undergraduate and graduate students in applied mathematics, physics, engineering, and chemistry in the exploration of rare events, multiscale processes and their analysis and simulation.The PI is interested in the large deviations regime (tail events) as well as in the moderate deviations regime (the gap between the typical center and the tail of the distribution). Due to the lack of explicit solutions, one has to rely on approximation and simulation methods and for this reason rigorous development of provably efficient approximation methods and simulation Monte Carlo methods is essential. In a closely related direction, the PI develops a rigorous theory of metastability for infinite dimensional dynamical systems that may have multiple scales, interacting with the rare events of interest. Moreover, the PI is interested in the effect of multiple scales on tail events associated to interacting particle systems. Systems of interacting diffusions arise in many areas of science, theory of random matrices, mathematical biology, neural networks in machine learning and optimization, construction of Kahler-Einstein metrics, opinion dynamics, finance, and engineering. The proposed work leads to the development of a rigorous mathematical framework that allows to design provably efficient algorithms associated to modeling of rare events. It will crystalize concepts and methods that are not well understood such as the effect of metastability on Monte Carlo methods and of multiple scales on large deviations and Monte Carlo methods for stochastic dynamical systems in infinite dimensions and for interacting particle systems.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.