Branching processes, partial differential equations and applications to knowledge diffusion

Dynamics in heterogeneous and random environments arise in a broad range of scientific fields, from economic growth and fluid turbulence to diffusion models in machine learning. These systems are often modelled by nonlinear partial differential equations and stochastic processes involving multiple temporal and spatial scales. To mitigate this complexity, researchers frequently use approximate macroscopic effective models or simplified systems that inherit essential qualitative properties of the dynamics of the larger system. This project seeks to study the validity and behavior of such approximations in applications across the physical sciences, macroeconomics, and machine learning. The project also contributes to the training and development of junior researchers in active areas of applied mathematics. The research is structured around three main themes. The first part of the project develops new tools and better understanding of the connection between branching processes, mean field games, and knowledge diffusion in macroeconomic models, with particular attention to their long-time behavior. The second part concerns the mathematical foundations of diffusion models used in machine learning, focusing on their convergence rates and regimes of validity in sampling and generative tasks. The third part investigates the long-time behavior of the solutions to systems of reaction-diffusion equations. Together, these efforts aim to advance both theoretical insight and practical methodologies in modeling systems influenced by randomness and heterogeneity. 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.