Scaling limits of spatial stochastic differential equations

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One central subject of modern probability research is to analyze large random systems and describe the expected dynamics and fluctuations. The goal of this research program is to study these descriptions via scaling limits of stochastic differential equations in the spatial setting. In addition to time, points in spatial structures such as discrete graphs or Euclidean spaces parameterize these stochastic differential equations. The main projects investigate stochastic spatial populations and interface growth models. The training of HQP will involve both of these two directions.   1. Diffusion processes for large spatial populations. This direction continues our previous study of a spatial death--birth process, known as the voter model, and its weak perturbations. The original problem for those results arises from theoretical biology; the results prove mean--field properties on general spatial structures in the form of convergences to diffusion processes. The current projects continue to consider questions from the biological literature and at the frontier of probability theory. We investigate more delicate scaling limits of the previous models. A more important goal is to establish scaling limits of other models as non--weak perturbations of the voter model. The study for these non--weak perturbations will begin with extending related heuristics of Aldous and Durrett to the spatial setting. In all cases, the methods will include diffusion theory and tools for mixing and metastability of Markov chains. The direction is expected to involve super--Brownian motion or more general superprocesses. These mathematical objects are given by scaling limits of the closely related branching processes on integer lattices. 2. Gaussian fluctuations in two--dimensional surface growth models. The main goal of this direction is to study Wolf's conjecture for the anisotropic Kardar--Parisi--Zhang (KPZ) equation. In this framework, stochastic partial differential equations physically describe scaling limits of surface growth models. As in the current progress of this area, the projects investigate scaling limits of particular models. They will be approached using diffusion theory and techniques for Gaussian distributions, including Fourier analysis for Gaussian free fields and Malliavin calculus. The results will extend our understanding of universality in Wolf's conjecture. In the physics literature, models of the complementary isotropic class feature non--Gaussian fluctuations. To obtain appropriate experience for non--Gaussian behavior, the proposal will extend to the study of the Airy line ensembles and spin glass models. The study of the Airy line ensembles will be approached using probabilistic methods for Brownian motions as in the work of Corwin and Hammond. Spin glass models are essential in statistical physics and theoretical computer science so that the study is of independent interest.