Singular stochastic PDEs and related statistical physics models

奇异随机偏微分方程和相关统计物理模型

• 批准号: EP/N021568/2
• 负责人: Weijun Xu
• 金额: $ 23.58万
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FN021568%2F2
• 关键词: Singular; stochastic; PDEs; related; statistical

The context of the proposal mainly concerns singular stochastic PDEs and related statistical physics models. By saying singular, we mean that the solution (or some of its derivatives) has wild oscillations with a frequency and magnitude blowing up to infinity at small scales. The singularities in the solutions to stochastic PDEs are typically almost everywhere. As a consequence, nonlinear operations of the solutions may not make sense as they take these high frequency oscillations into quantities that are typically infinity. Thus, the correct interpretation of the solutions to these equations usually requires renormalisation. In the past three years, there have been major advances in the development of solution theories to a number of important singular SPDEs, including the three dimensional stochastic quantisation equation, the KPZ equation and the parabolic Anderson model in two and three dimensions. These equations are widely believed to be the universal models for the large scale behaviours of many systems in statistical mechanics. The successful construction of the solutions opens a way to study in detail these equations as well as the natural phenomena they represent. In this proposal, we aim to deepen the understanding of the quantitative behaviour of the solutions to these equations, and rigorously prove the universality phenomena for their related statistical physics models. We will also investigate how certain perturbations of the system (for example, asymmetry in phase coexistence models) can force its large scale behaviour to deviate from the expected universal limit.

本文主要研究奇异随机偏微分方程及其相关的统计物理模型。说到奇异，我们的意思是解（或它的一些导数）在小尺度下具有频率和幅度膨胀到无穷大的疯狂振荡。随机偏微分方程解中的奇异性几乎无处不在。因此，解的非线性运算可能没有意义，因为它们将这些高频振荡变为无穷大的量。因此，这些方程的解的正确解释通常需要重整化。在过去的三年中，一些重要的奇异SPDEs的解理论的发展取得了重大进展，包括三维随机量化方程、KPZ方程和二维和三维的抛物型安德森模型。这些方程被广泛认为是统计力学中许多系统大尺度行为的通用模型。解的成功构造为详细研究这些方程及其所代表的自然现象开辟了一条道路。在本文中，我们旨在加深对这些方程解的定量行为的理解，并严格证明其相关统计物理模型的普遍性现象。我们还将研究系统的某些扰动（例如，相共存模型中的不对称）如何迫使其大尺度行为偏离预期的普遍极限。

项目成果

Sharp Convergence of Nonlinear Functionals of a Class of Gaussian Random Fields.

一类高斯随机场的非线性泛函的急剧收敛。

• DOI: 10.1007/s40304-018-0162-9
• 发表时间: 2018
• 期刊: Communications in mathematics and statistics
• 影响因子: 0.9
• 作者: Xu W