Large scales analysis of SPDEs

近十年来，对KPZ方程和KPZ不动点的研究取得了显著进展。自Kardar、Parisi和Zhang [KPZ86]于1986年提出KPZ方程以来，它一直是捕获各种离散物理模型和随机界面生长动力学的默认模型。该方程在维数d=1上已经是病态的，是奇异SPDEs的主要例子之一。由于噪声的不规则性和方程中涉及的非线性项，这类非线性SPDEs一直是难以解决的问题。直到Hairer的开创性工作[H13]和他随后对正则结构的发展，现在提供了一个健壮的框架来研究几乎所有类型的局部次临界奇异spde，允许通过适当的重整化来理解方程及其解。与此同时，最近的各种努力最终得到了KPZ不动点的精确描述，并证明了KPZ解决方案的大规模收敛性。另一方面，大尺度涨落的研究最近也在高维d >= 2中取得了各种进展[CSZ20， MU18]，其中爱德华兹-威尔金森（高斯）通适性类是弱无序状态中的吸引不动点。在这里（临界/超临界设置），规则结构和副控制分布的先驱理论不再适用，因此，为了使研究集中在驱动白噪声适当正则化的方程有意义。目的和目标：通常这种大规模的分析已经考虑了高斯驱动噪声在微观水平上具有有限范围的相关性（来自紧密支持的mollifiers）。我们想研究噪声的长期相关性（无论是在时间或空间上，还是联合）对大尺度动力学/统计学的影响。因此，了解是否在大尺度上显示相同的普用性类行为，并了解是否可能根据噪声相关衰减和空间维度d发生相变。方法的新颖性：目前，对KPZ方程有一些理解和期望，这些理解和期望来自数值模拟和物理文献（d <= 2），这些观点有时指向相反的方向，缺乏完全的数学处理。因此，将需要新的方法来研究分析的长期相关制度，超越现有数学文献中的短期相关设置。该项目与EPSRC的以下研究领域保持一致：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., Zhang, yc ., 1986。生长界面的动态缩放。物理评论通讯，56(9),p.889。[H13]王晓明，陈晓明，2013。解KPZ方程。《数学年鉴》第559-664页。[CSZ20]张建军，张建军，张建军，等。整个亚临界区域的二维KPZ方程。概率论，48(3)，页1086-1127。[MU18]李建军，李建军，李建军，2018。KPZ方程在空间维度3及以上的标度极限。统计物理学报，21 (4),pp. 563 -598

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