Studies of the Stochastic Partial Differential Equations

The work will study important properties of solutions to stochastic partial differential equations (SPDEs). These equations have applications in many fields, ranging from physics and chemistry to biology and financial markets. Key examples include the stochastic heat equation (SHE), the parabolic Anderson model (PAM), and the Kardar-Parisi-Zhang (KPZ) equation. These equations and their variants model crucial phenomena such as the spread of forest fires, smoke dispersal, growth of tumor tissues, superconductivity, and the formation of the universe's stratified structure and magnetic fields of planets or stars. This project will deepen our understanding of these phenomena by establishing essential properties of solutions, such as existence and uniqueness, moment asymptotics, positivity and support of solutions, existence and smoothness of probability density. Active efforts will be made to encourage participation by students from underrepresented groups in this research program, fostering an inclusive and diverse research environment. The investigator also participates in outreach programs for pre-college students. SPDEs depend on many technical parameters and conditions, including initial conditions, noise structures, conditions on the diffusion coefficients, geometric and analytic properties of the spatial domain and the corresponding boundary conditions, differential operators, among others. This project will focus on: (1) Studying SHE/PAM with rough initial conditions; (2) Investigating SHE in the sublinear growth regime, possibly with non-Lipschitz diffusion coefficients; and (3) Exploring the boundary effects to SHE/PAM. These three objectives will be implemented via six concrete projects. This project is jointly funded by the DMS Probability program and the Established Program to Stimulate Competitive Research (EPSCoR). 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.