Stochastic partial differential equations: the critical dimension and invariant measures

Stochastic partial differential equations (SPDEs) are models of space-time systems subject to random effects. They are especially useful in studying systems in which the randomness is rapidly decorrelating in space and/or time, meaning that different parts of the system are subject to independent random noise. Mathematical studies of such systems are often concentrated on how this small-scale randomness manifests in larger-scale properties of the system. The present project is aimed at studying two aspects of the theory of SPDEs. The first aspect concerns SPDEs in critical dimensions. These SPDEs have a scale-invariance that puts them just beyond the scope of many general techniques but also gives them a symmetry that can be leveraged for computations. The second aspect is to study the long-time behavior of SPDEs on unbounded domains. This involves questions about the relationship between "long-time" and "large-space" effects in random systems. The awardee also plans to engage in mentoring and exposition to broaden the use and understanding of probabilistic techniques. One goal of the present project is to work towards a general theory of SPDEs in the critical dimension. The study of stochastic heat equations in dimension two has yielded a rich set of phenomena, and the awardee plans to develop ways to understand these phenomena for larger classes of equations, such as stochastic heat and Hamilton–Jacobi equations with more general nonlinearities. The awardee also plans to further develop the understanding of critical-dimension/critical-temperature problems such as the Critical Stochastic Heat Flow. Another goal is to work towards qualitative understanding of SPDE invariant measures that do not admit explicit descriptions, and to use this understanding to study additive functionals of these processes. 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.