Scaling limits of spatial stochastic differential equations

现代概率论研究的一个中心课题是分析大型随机系统并描述预期的动态和波动。本研究计划的目的是研究这些描述通过尺度限制的随机微分方程在空间设置。除了时间之外，离散图或欧几里得空间等空间结构中的点也将这些随机微分方程参数化。主要项目研究随机空间种群和界面增长模型。HQP的培训将涉及这两个方向。1. 大空间种群的扩散过程。这个方向继续了我们之前对空间死亡-出生过程的研究，被称为选民模型，以及它的弱扰动。这些结果的最初问题来自理论生物学；结果证明了在一般空间结构上以扩散过程收敛形式的平均场性质。目前的项目继续考虑来自生物学文献和概率论前沿的问题。我们研究了先前模型的更精细的缩放极限。一个更重要的目标是建立其他模型的尺度限制作为选民模型的非弱扰动。对这些非弱扰动的研究将从将Aldous和Durrett的相关启发式扩展到空间环境开始。在所有情况下，方法将包括扩散理论和马尔可夫链的混合和亚稳态的工具。该方向预计将涉及超布朗运动或更一般的超过程。这些数学对象是由整数格上密切相关分支过程的尺度极限给出的。2. 二维表面生长模型中的高斯波动。本方向的主要目标是研究各向异性Kardar—Parisi—Zhang （KPZ）方程的Wolf猜想。在这个框架中，随机偏微分方程物理地描述了表面生长模型的缩放极限。在该领域的当前进展中，这些项目研究特定模型的缩放限制。他们将使用高斯分布的扩散理论和技术，包括高斯自由场的傅立叶分析和马利文微积分。这些结果将扩展我们对沃尔夫猜想的普遍性的理解。在物理文献中，互补各向同性类的模型具有非高斯波动的特征。为了获得非高斯行为的适当经验，该建议将扩展到艾里线系综和自旋玻璃模型的研究。艾里线系综的研究将采用布朗运动的概率方法，就像考文和哈蒙德的工作一样。自旋玻璃模型在统计物理和理论计算机科学中是必不可少的，因此研究是独立的兴趣。

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.