Analysis of Stochastic Partial Differential Equations

This project is chiefly concerned with the development of new mathematical methods in the rapidly growing field of stochastic partial differential equations (SPDEs). Special attention is paid to developing novel techniques that are designed to analyze specific families of SPDEs that arise naturally in application areas that range from mathematical oceanography and climate models, stochastic hydrology, geostatistics, classical cosmology, to statistical and mathematical physics. Although many of the problems studied as part of the project are centered around noteworthy questions in SPDEs, a successful resolution of these problems will likely also help study other complex systems. The developed ideas and techniques are expected to have sufficient novelty to open new research areas, solve a number of long-standing open problems and promote further applicability of the theory of SPDEs. The project will involve graduate students, and the PI will continue organizing conferences and is planning to co-author a textbook in the area of research. The project will engage quantitative connections between SPDEs, random fields, and random fractals in the context of concrete questions of independent, modern interest. Some of these connections are motivated directly by questions in applications areas. These include problems that range from nonlinear statistical inverse problems to the analysis of the blowup phenomenon for noisy reaction-diffusion equations. Others are aimed at providing mathematical explanations for physically observed phenomena such as intermittency and void dynamics. Yet others are related to studying entirely new phenomena such as macroscopic and microscopic multifractality. This research is expected to yield novel insights into the structure of SPDEs, blowup phenomena, physical multifractals, and the related multifractal random fields. 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.