Nonlinear Stochastic Partial Differential Equations and Applications

Statistical uncertainty plays a significant role in a diverse range of models for complicated dynamic phenomena, leading to wild, stochastic behavior. Such probabilistic effects are caused, for instance, by unpredictable market shifts in the global economy, or turbulent or chaotic weather patterns at the evolving front of a massive forest fire. The investigator will develop a mathematical understanding for the equations arising in these applications, while also studying the stabilizing and regularizing effects of stochastic noise, for which there is often experimental or numerical evidence. This project will generate opportunities to mentor graduate and undergraduate students by providing both professional advice and mathematical knowledge related to the project. Dynamical random behavior under various complex influences is often described by nonlinear stochastic partial differential equations. Such equations cannot be solved through the superposition of simple formulae and are therefore not yet well-understood mathematically. The project will draw on tools from functional analysis and probability to resolve the well-posedness of nonlinear stochastic partial differential equations arising in competitive large population dynamics and in stochastically forced interface evolutions. The effects of stochasticity will be further analyzed by studying the long-time behavior of solutions, probabilistic averaging and regularizing phenomena, and stochastic selection principles for models with a small level of background noise. The material influence of stochasticity indicates that the statistical fluctuations in experimental data cannot be completely ignored, thereby justifying the technical study of those stochastic partial differential equations involved in this project. 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.