Scaling limits of spatial stochastic differential equations

现代概率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。这个研究项目的目的是通过空间环境中随机微分方程的尺度极限来研究这些描述。除了时间，空间结构中的点，如离散图或欧几里德空间，还将这些随机微分方程式参数化。主要项目研究随机空间种群和界面生长模型。HQP的训练将涉及这两个方向。1.大空间种群的扩散过程。这个方向延续了我们之前对空间死亡-出生过程的研究，也就是所谓的选民模型，以及它的弱扰动。这些结果的原始问题来自理论生物学；结果证明了一般空间结构的平均场性质是以收敛于扩散过程的形式出现的。目前的项目继续考虑生物学文献中的问题和概率论的前沿问题。我们研究了以前模型的更精细的比例限制。一个更重要的目标是建立其他模型的比例极限，作为投票者模型的非弱扰动。对于这些非弱扰动的研究将从将Aldous和Durrett的相关启发式推广到空间环境开始。在所有情况下，这些方法都将包括扩散理论和马尔科夫链混合和亚稳性的工具。这个方向预计将涉及超布朗运动或更一般的超过程。这些数学对象是通过整数格上密切相关的分支过程的尺度极限来给出的。2.二维表面生长模型中的高斯涨落。这个方向的主要目的是研究各向异性Kardar-Parisi-Zhang(KPZ)方程的Wolf猜想。在这个框架中，随机偏微分方程组物理地描述了表面生长模型的尺度极限。与这一领域目前的进展一样，这些项目调查特定模型的比例限制。他们将使用扩散理论和高斯分布的技术，包括高斯自由场的傅里叶分析和Malliavin微积分。这些结果将扩展我们对沃尔夫猜想中普适性的理解。在物理学文献中，互补各向同性类的模型以非高斯涨落为特征。为了获得非高斯行为的适当经验，该提议将扩展到对艾里线系综和自旋玻璃模型的研究。艾里线系综的研究将使用布朗运动的概率方法，就像在Corwin和Hammond的工作中一样。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，因此这项研究是独立的。

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.