Exact solvability of the Kardar-Parisi-Zhang stochastic partial differential equation

Kardar-Parisi-Zhang（KPZ）方程是一个非线性随机偏微分方程，其统计性质被认为描述了一大类数学模型，包括相互作用粒子系统，随机增长模型，定向聚合物和分支扩散过程。该项目的目的是取得重大进展，发展一个理论的精确解的KPZ方程。特别是，这一理论应导致精确的公式的概率分布的解决方案KPZ方程时，开始与各种重要类型的初始数据。这个项目将涉及一些数学领域，这个方向已经产生了这些领域独立感兴趣的结果，其中包括：麦克唐纳对称函数理论，热带组合数学和惠特克函数，以及某些量子可积系统。自从两百年前发现高斯分布（钟形曲线）以来，它已经成为数学最伟大的社会和科学贡献之一。这是一个强大的理论，解释和分析了世界上许多固有的随机性。用高斯统计精确描述的物理和数学系统被称为高斯普适类。然而，这个类并不是包罗万象的。例如，经典的极值统计或泊松统计可以更好地捕捉从自然灾害到急诊室访问等事件的随机性和严重性。最近，重要的研究工作一直集中在理解系统，没有很好地描述在任何经典的发展统计。这些系统的失败，以符合经典的描述，一般是由于自然观测和随机输入和噪声的潜在来源之间的非线性关系。在数学、物理学、材料科学、化学和生物学领域，人们已经积极研究了各种复杂系统的模型，如生长过程、聚合物链、质量传输、交通流、扩散理论、驱动气体和湍流。所有这些系统都不符合经典的高斯统计，如已经通过实验证据观察到的，涉及湍流液晶，薄膜上的晶体生长，刻面边界，细菌菌落生长，纸张润湿，裂纹形成和燃烧前沿。令人惊讶的是，尽管它们存在差异，但所有的系统都属于一个新的统计普适类，其属性都是用一个称为Kardar-Parisi-Zhang（KPZ）方程的模型来描述的。这个项目的目的是发展KPZ方程及其普适类的统计理解。

The Kardar-Parisi-Zhang (KPZ) equation is a non-linear stochastic partial differential equation whose statistical properties are believed to describe a large class of mathematical models including interacting particle systems, random growth models, directed polymers, and branching diffusion processes. The purpose of this project is to make significant progress towards developing a theory of the exact solvability of the KPZ equation. In particular, this theory should lead to exact formulas for the probability distributions of the solution to the KPZ equation when started with various important types of initial data. This project will involve a number of fields of mathematics and this direction has already produced results of independent interest to these fields, which include: Macdonald symmetric function theory, tropical combinatorics and Whittaker functions, and certain quantum integrable systems. Since its discovery two hundred years ago the Gaussian distribution (bell curve) has come to represent one of mathematics greatest societal and scientific contributions ? a robust theory explaining and analyzing much of the randomness inherent in the world. Physical and mathematical systems accurately described in terms of Gaussian statistics are said to be in the Gaussian universality class. This class, however, is not all encompassing. For example, classical extreme value statistics or Poisson statistics better capture the randomness and severity of events ranging from natural disasters to emergency room visits. More recently, significant research efforts have been focused on understanding systems which are not well-described in terms of any of the classically developed statistics. The failure of these systems to conform to classical descriptions is generally due to a non-linear relationship between natural observables and underlying sources of random inputs and noise. A variety of models for complex systems such as growth processes, polymer chains, mass transport, traffic flow, queueing theory, driven gases, and turbulence have been actively studied for over forty years in mathematics, physics, material science, chemistry and biology. All of these systems fail to conform with classical Gaussian statistics, as has been observed through experimental evidence involving turbulent liquid crystals, crystal growth on a thin film, facet boundaries, bacteria colony growth, paper wetting, crack formation, and burning fronts. Surprisingly, despite their differences, all of the systems fall into a new statistical universality class whose properties are described in terms of a single model called the Kardar-Parisi-Zhang (KPZ) equation. The purpose of this project is to develop a statistical understand of the KPZ equation and its universality class.