Large scales analysis of SPDEs

在过去的十年里，对KPZ方程和KPZ不动点的研究取得了显著的进展。KPZ方程自1986年由Kardar，Parisi和Zhang[KPZ86]提出以来，一直是捕捉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d=1已经是不适定的，这是奇异SPDEs的主要例子之一。由于方程中噪声和非线性项的不规则性，这种类型的非线性SPDEs长期以来一直是难以处理的。直到海尔[H13]的开创性工作[H13]和他随后发展的正则性结构，现在提供了一个健壮的框架来研究几乎所有类型的局部亚临界奇异SPDEs，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解的大范围收敛。另一方面，最近对大尺度涨落的研究也在高维d&gt；=2方面取得了各种进展[CSZ20，MU18]，其中Edwards-Wilkinson(Gauss)普适性类是弱无序状态下吸引人的不动点。在这里(临界/超临界设置)，正则性结构和准受控分布的开创性理论不再容易应用，因此，为了使研究集中的方程具有适当的正则化，驱动白噪声得到了合理的解释。目的和目的：通常，这种大尺度分析考虑了具有有限范围相关性的微观层面的高斯驾驶噪声(来自紧凑支承的软化器)。我们想要研究噪声的长程相关性(无论是在时间上还是在空间上，或者联合地)对大尺度动力学/统计的影响。因此，理解是否在大尺度上显示相同的普适性类行为，并理解是否可能发生相变，取决于噪声相关性衰减和空间维度d。方法的新颖性：目前对KPZ方程有一些理解和期望，来自数值模拟和来自物理文献的工作(在d&lt；=2中)，这些点有时指向相反的方向，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性的长程关联机制，超越现有数学文献中的短程关联设置。该项目与EPSRC的以下研究领域相一致：数学分析、数学物理、统计学和应用概率。参考文献：Kardar，M.，Parisi，G.和Zhang，Y.C.，1986。动态扩展不断增长的接口。《物理评论通讯》，第56(9)页，第889页。[H13]海尔，M.，2013。求解KPZ方程。数学年鉴，第559-664页。[CSZ20]Carvenna，F.，Sun，R.和Zygouras，N.，2020。整个亚临界区域的二维KPZ方程。《概率年鉴》，48(3)，1086-1127页[MU18]Magnen，J.和Unterberger，J.KPZ方程在3维及更高空间中的标度极限。《统计物理学报》171(4)，第543-598页

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