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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=718349
• 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。本研究计划的目标是通过在空间设置的随机微分方程的尺度限制来研究这些描述。除了时间之外，空间结构中的点，如离散图或欧几里得空间，也是这些随机微分方程的参数。 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。 这个方向继续我们以前的研究空间死亡-出生过程，被称为选民模型，和它的弱扰动。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收敛扩散过程的形式的平均场性质。 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。我们调查更微妙的缩放限制以前的模型。一个更重要的目标是建立其他模型的标度限制作为非弱扰动的选民模型。对这些非弱扰动的研究开始时将把Aldous和Durrett的相关理论推广到空间环境。在所有情况下，方法将包括扩散理论和工具的混合和亚稳定的马尔可夫链。 该方向预计将涉及超布朗运动或更一般的超过程。这些数学对象由整数格上密切相关的分支过程的标度极限给出。 2.二维表面生长模型中的高斯涨落。 这个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描述的表面生长模型的尺度限制。正如该领域目前的进展一样，这些项目调查了特定模型的缩放限制。他们将接近使用扩散理论和技术的高斯分布，包括傅立叶分析高斯自由场和Malliavin演算。这些结果将扩展我们对Wolf猜想普适性的理解。 在物理学文献中，互补各向同性类的模型具有非高斯波动的特征。为了获得适当的经验，非高斯行为，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，如Corwin和哈蒙德的工作。自旋玻璃模型在统计物理学和理论计算机科学中是必不可少的，因此该研究具有独立的兴趣。