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

在过去的十年中，对KPZ方程和KPZ固定点的研究已经取得了显着进步。自1986年由Kardar，Parisi和Zhang [KPZ86]引入以来，KPZ方程一直是捕获各种离散物理模型和随机接口增长的动态的默认模型。该方程式已经不足D = 1，是奇异SPDE的主要例子之一。由于噪声和方程中涉及的非线性项的不规则性，这种类型的非线性SPD长期很难进行。这要直到毛发师[H13]的开创性工作及其随后的规律性结构的发展，现在，它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SPDES，从而可以通过适当的重新分配来理解方程式及其解决方案。同时，最近的各种努力最终以KPZ固定点的精确描述，并证明了KPZ解决方案的较大尺度收敛。另一方面，对大尺度波动的研究最近在较高的维度d> = 2中看到了各种进展[CSZ20，MU18]，其中Edwards-Wilkinson（Gaussian）普遍性类别是弱点障碍制度中吸引的固定点。在这里（临界/超临界设置）规律性结构和副控制分布的开创性理论不再容易应用，因此要理解研究的方程式，该方程是为了驱动白噪声适当正规化。目的和目标：通常，这种较大的量表分析考虑了高斯驱动噪声，在显微镜水平上具有有限的范围相关性（来自紧凑的支撑Mollifiers）。我们想研究噪声的远距离相关性（在时间或空间，或共同的）对大尺度动力学/统计数据的影响。因此，了解是否在大尺度上显示相同的普遍性类行为，并了解是否会发生相变，具体取决于噪声相关性衰减和空间维度d。方法论的新颖性：目前，对来自数值模拟的KPZ方程和物理学文献的作品（在d <= 2中）的作品有一些理解和期望，有时在对比方向上这些点，并且缺乏完全的数学处理。因此，将需要新的方法来研究分析长范围的相关性制度，超出了现有数学文献中的短范围相关设置。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。物理评论信，56（9），第889页。[H13] Hairer，M.，2013。解决KPZ方程。数学年鉴，第559-664页。[CSZ20] Caravenna，F.，Sun，R。和N.概率年鉴，48（3），第1086-1127页。[MU18] Magnen，J。和Unterberger，J。，2018年。空间维度3及更高的KPZ方程的缩放限制。统计物理学杂志，171（4），pp.543-598