CAREER: Stochastic Partial Differential Equations: Analysis and Applications

About This Grant

The research project focuses on stochastic partial differential equations (SPDEs), which play a significant role in explaining complex phenomena across diverse fields such as physics, biology, and chemistry; they can model real-world problems, including tumor growth, forest fire propagation, superconductivity, and the structure of the universe. The broader impacts of this project involve enhancing societal well-being through potential advancements in cancer treatment, fire management, environmental processes such as sediment transport, and improving manufacturing processes in pharmaceuticals and semiconductors. The project also promotes interdisciplinary applications of SPDEs in finance, numerical analysis, engineering, and machine learning, and provides research opportunities for graduate students. Additionally, the researchers will foster educational outreach through the continued development of open-access computational tools, comprehensive mathematical bibliographic databases, specialized analytical tools, and STEM education programs aimed at middle and high school students. This project addresses critical challenges in SPDE theory by investigating the stochastic heat equation (SHE), the parabolic Anderson model, and the Kardar-Parisi-Zhang equation under conditions of rough initial data and diverse noise structures. A significant innovation is demonstrating the existence of nontrivial invariant measures for SHE/PAM resulting from interactions between noise structures and rough initial data. The project also seeks to resolve prominent open problems related to the moment estimates of the Malliavin derivative, thus expanding the applicability of Malliavin calculus to SPDEs. By studying fractional SPDEs, the researchers aim to develop a unified theoretical framework interpolating among various properties, including the sample path regularities of SHE and the stochastic wave equation. Complementing theoretical developments, realistic simulations involving inhomogeneous particles and long-range correlations will be performed, aiming to produce novel conjectures and deeper insights into stochastic/disordered systems. 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.