Large scales analysis of SPDEs

SPDE 的大规模分析

• 批准号: 2442362
• 金额: --
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2442362
• 关键词: Large; scales; analysis; SPDEs

The study of the KPZ equation and the KPZ fixed point have witnessed remarkable progresses in the last decade. Since its introduction by Kardar, Parisi and Zhang [KPZ86] in 1986, the KPZ equation has been the default model to capture the dynamics of a large variety of discrete physical models and random interfaces growth. The equation is ill-posed already in dimension d=1, being one of the main examples of singular SPDEs. This type of nonlinear SPDEs for long had been intractable due to the irregularity of the noise and nonlinear terms involved in the equations. This until the seminal work of Hairer [H13] and his subsequent development of Regularity Structures, which now provide a robust framework to study virtually all type of locally subcritical singular SPDEs, allowing to make sense of the equations and their solution through appropriate renormalization. In the meantime various efforts recently culminated with an exact description of KPZ fixed point and proved the large scales convergence of the KPZ solution. On the other hand, the study of large scales fluctuations has recently seen various progresses [CSZ20, MU18] also in higher dimensions d >= 2, where the Edwards-Wilkinson (Gaussian) universality class is the attracting fixed point in weak disorder regimes. Here (critical/super-critical settings) the pioneering theories of Regularity Structures and Paracontrolled Distributions no longer readily apply, hence to make sense of the equations the study has focused with driving white noise appropriately regularized. Aims and Objectives:Typically this large scales analysis has considered Gaussian driving noise with finite range correlations at the microscopic level (coming from compactly supported mollifiers). We want to investigate the impact of long range correlations of the noise (either in time or space, or jointly) on the large scales dynamics/statistics. Hence understand if the same universality class behaviour is displayed at large scales, and understand whether there may occur phase transitions depending on the noise correlations decay and the spatial dimension d. Novelty of the methodology:At present there is some understanding and expectations for the KPZ equation coming from numerical simulations and works from the physics literature (in d <= 2), these point in contrasting directions at times and lack a fully mathematical treatment. Hence new methodologies will be required to investigate analytically long range correlations regimes, going beyond the short range correlations settings in the existing mathematical literature. The project is aligned with the following EPSRC research areas: Mathematical Analysis, Mathematical Physics, Statistics and Applied Probability. References:[KPZ86] Kardar, M., Parisi, G. and Zhang, Y.C., 1986. Dynamic scaling of growing interfaces. Physical Review Letters, 56(9), p.889.[H13] Hairer, M., 2013. Solving the KPZ equation. Annals of mathematics, pp.559-664.[CSZ20] Caravenna, F., Sun, R. and Zygouras, N., 2020. The two-dimensional KPZ equation in the entire subcritical regime. The Annals of Probability, 48(3), pp.1086-1127.[MU18] Magnen, J. and Unterberger, J., 2018. The scaling limit of the KPZ equation in space dimension 3 and higher. Journal of Statistical Physics, 171(4), pp.543-598

近十年来，KPZ方程和KPZ不动点的研究取得了显着的进展。自 Kardar、Parisi 和Zhang [KPZ86] 于 1986 年提出以来，KPZ 方程一直是捕获各种离散物理模型和随机界面增长动力学的默认模型。该方程在 d=1 维上已经不适定，是奇异 SPDE 的主要例子之一。由于方程中涉及的噪声和非线性项的不规则性，这种类型的非线性 SPDE 长期以来一直很棘手。这直到 Hairer [H13] 的开创性工作以及他随后对正则结构的发展，现在提供了一个强大的框架来研究几乎所有类型的局部亚临界奇异 SPDE，允许通过适当的重正化来理解方程及其解。与此同时，最近的各种努力最终达到了 KPZ 不动点的精确描述，并证明了 KPZ 解的大规模收敛性。另一方面，大尺度波动的研究最近在更高维度 d >= 2 中也取得了各种进展 [CSZ20，MU18]，其中爱德华兹-威尔金森（高斯）普适性类是弱无序状态中吸引人的固定点。在这里（临界/超临界设置），正则结构和副控制分布的开创性理论不再容易应用，因此为了理解方程的意义，研究重点是驱动白噪声适当正则化。目的和目标：通常，这种大规模分析考虑了微观水平上具有有限范围相关性的高斯驱动噪声（来自紧凑支撑的缓和器）。我们想要研究噪声的长程相关性（时间或空间，或联合）对大规模动态/统计的影响。因此，了解是否在大尺度上显示相同的普遍性类行为，并了解是否可能发生取决于噪声相关性衰减和空间维度 d 的相变。方法论的新颖性：目前，对 KPZ 方程的一些理解和期望来自于数值模拟和物理文献（d <= 2）中的工作，这些有时指向相反的方向，并且缺乏完整的数学处理。因此，需要新的方法来研究分析性的长期相关机制，超越现有数学文献中的短期相关设置。该项目与以下 EPSRC 研究领域保持一致：数学分析、数学物理、统计学和应用概率。参考文献：[KPZ86] Kardar, M., Parisi, G. 和Zhang, Y.C., 1986。生长界面的动态缩放。物理评论快报，56(9)，第 889 页。[H13] Hairer, M., 2013。求解 KPZ 方程。数学年鉴，第 559-664 页。[CSZ20] Caravenna, F.、Sun, R. 和 Zygouras, N., 2020。整个亚临界状态下的二维 KPZ 方程。 《概率年鉴》，48(3)，第 1086-1127 页。[MU18] Magnen, J. 和 Unterberger, J.，2018。空间维度 3 及更高维度中 KPZ 方程的缩放极限。统计物理学杂志，171(4)，第 543-598 页