Scaling limits of spatial stochastic differential equations

空间随机微分方程的标度极限

• 批准号: RGPIN-2020-06500
• 负责人: Chen, YuTing
• 金额: $ 1.68万
• 依托单位: University of Victoria
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758932
• 关键词: Scaling; limits; spatial; stochastic; differential

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.

现代概率研究的一个核心主题是分析大型随机系统并描述预期的动态和波动。该研究计划的目的是通过空间环境中随机微分方程的缩放限制来研究这些描述。除了时间外，空间结构（例如离散图或欧几里得空间）中的点可以参数化这些随机微分方程。主要项目研究了随机的空间人群和界面增长模型。 HQP的培训将涉及这两个方向。 1。大型空间种群的扩散过程。这个方向延续了我们先前对空间死亡的研究 - 出生过程，称为选民模型及其弱势扰动。这些结果的最初问题来自理论生物学。结果证明了均值 - 场在一般空间结构上的特性，以差异过程的融合形式。当前的项目继续考虑生物学文献和概率理论边界的问题。我们研究了先前模型的更精致的缩放限制。一个更重要的目标是建立其他模型的扩展限制，因为选民模型的非潜在扰动。这些非运动扰动的研究将从将Aldous和Durrett的相关启发式方法扩展到空间环境开始。在所有情况下，这些方法都将包括扩散理论和马尔可夫链混合和亚竞争力的工具。该方向有望涉及超级 - 布朗尼运动或更通用的超级过程。这些数学对象是通过在整数晶格上的紧密相关分支过程的缩放限制给出的。 2。二维表面生长模型中的高斯波动。这个方向的主要目标是研究狼的概念，以示各向异性的kardar-parisi-zhang（kpz）方程。在此框架中，随机部分微分方程物理描述了表面生长模型的缩放限制。就像在该领域的当前进展中一样，这些项目研究了特定模型的规模限制。将使用差异理论和高斯分布的技术来对其进行处理，包括高斯自由场和Malliavin colkulus的傅立叶分析。结果将扩展我们对沃尔夫概念普遍性的理解。在物理文献中，完整的各向同性类特征非高斯波动的模型。为了获得非高斯行为的适当经验，该提案将扩展到对通风线的合奏和自旋玻璃模型的研究。像科尔文和哈蒙德的工作一样，将使用有问题的布朗动议的问题方法对通风线的集合进行研究。自旋玻璃模型在统计物理学和理论计算机科学中至关重要，因此该研究具有独立的兴趣。