Large-Scale Behavior of Interacting Particle Systems

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Interacting particle systems form a mathematical framework for systems with interacting components that arise in a variety of fields in science and engineering. Examples include, among many others, financial institutions interacting with each other via borrowing and lending, synchronization of neurons in the brain via electrical signals, and physical particles interacting through the (e.g. electromagnetic) potential they themselves generate. In many situations, the systems involved are large, e.g. the number of neurons in a given part of the human brain is on the order of a million. The goal of the project is to advance the quantitative understanding of such large interacting particle systems, relying on stochastic and deterministic partial differential equations. The project will tackle equations describing the large scale behavior of interacting particle systems and the blowups arising thereby. More specifically, blowups in Stefan problems for parabolic partial differential equations governing the limiting particle density in particle systems interacting through the hitting times will be investigated by a combination of probabilistic and analytic techniques. In particular, the exact timing of blowups will be determined, and the existence and uniqueness of a suitable solution will be established beyond the time of the first blowup. In addition, Gaussian approximations for the hitting times in interacting particle systems will be derived using the limiting stochastic partial differential equations. Finally, solutions to the latter arising from interacting particle systems will be shown to coincide with the entropy solutions arising naturally in the analysis of 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.