Estimating Probabilities of Rare Events in Interacting Particle Systems

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Estimation of probabilities of rare events is an important topic in many different areas. For example, quantification of risk of extreme outcomes is of central concern in many problems in finance, economics, environmental science, geophysics, and engineering. The mathematical framework for studying such problems is given by the theory of large deviations which is concerned with the characterization of decay rate of probabilities of deviations of a system with uncertainties from its nominal expected behavior. In the last twenty years a new approach to the study of large deviations problems, that brings to bear techniques from the theory of stochastic control, has become prominent. The goal of this research is to develop a systematic framework based on stochastic control ideas for an important and challenging class of large deviation problems that arise in the study of interacting particle systems. Interacting particle systems considered here are motivated by phenomena in Natural Sciences. Some examples include models for chemotaxis of biological particles, reaction-diffusion systems arising from ecological models, and crystal growth models from chemistry. In addition to developing the mathematical machinery for providing asymptotic bounds on decay rate of rare event probabilities, this work will develop accelerated Monte-Carlo methods for estimating probabilities of interest using methods of importance sampling that are inspired by a large deviation analysis of the underlying systems.This project will study four different families of particle systems: (A) Weakly interacting diffusions with a small common noise; (B) Microscopic particle models for Patlak-Keller-Segel equations; (C) Brownian particle systems for reaction-diffusion equations; (D) Locally interacting jump-diffusions. Under topic (A), the focus will be on large deviation asymptotics as the number of particles becomes large and the intensity of the common noise becomes small. The goal is to characterize different forms of large deviation behavior as the two parameters approach limits at different relative rates. Patlak-Keller-Segal equations in topic (B) are nonlinear non-local PDE that model active chemotaxis of biological particles. Interacting diffusive particle systems that are fully coupled with the evolution of the underlying chemical field have been used to give a mesoscopic description of the phenomenon. Goal of the proposed research here is to study large deviation problems aimed at understanding the long time behavior of such particle systems. In particular, this work will study invariant measure asymptotics and metastability behavior near a rest point. Under topic (C), large deviations behavior of particle systems of reaction-diffusion type will be studied. Of particular interest are annihilating Brownian particles approximating reaction diffusion equations with a polynomial reaction term in which the interaction becomes singular in the limit. Finally topic (D) is concerned with locally interacting particle systems on discrete lattices with a suitable temporal and spatial scaling. Such systems arise, for example, as crystal growth models in chemistry. The goal is to develop stochastic control methods for studying large deviation properties of such 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.