Branching Processes, Random Partial Differential Equations and Applications

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Physical, biological, and economic dynamics systems in heterogeneous and random environments are ubiquitous in nature, and include examples as diverse as spreading in ecology, economic growth, and fluid turbulence. The mathematical modeling of such systems involves nonlinear partial differential equations and stochastic processes, which for real-world applications contain a multitude of temporal and spatial scales, making the numerical simulation of the microscopic details of their behavior beyond reach even of modern computers. To mitigate this issue, one approach is to use approximate macroscopic effective models. Another is to consider simplified models that inherit the qualitative properties of the dynamics of the larger system. The overarching goal of this project is to study the validity of such approximations in applications of interest to physical sciences and macroeconomics, among others. The mentoring and training of graduate students will be integrated in the project.The first part of the project aims to develop new tools and better understanding of the connection between branching processes and nonlinear parabolic equations. One focus is on the study of branching Brownian motion in dimensions higher than one. The second part concerns the long-time behavior of nonlinear equations arising in the fluid dynamics and reaction-diffusion modeling, with a focus on very long-time scales, when fluctuations in the solutions start building up in ways beyond the classical central limit theorem time scales. The third part investigates the long-time behavior of the solutions to systems of mean-field equations that arise in macroeconomics.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.