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，包括相互作用的粒子系统、随机生长模型、定向聚合物和支化扩散过程。这个项目的目的是在发展KPZ方程的精确可解性理论方面取得重大进展。特别是，当从各种重要的初始数据类型开始时，这一理论应该导致KPZ方程解的概率分布的精确公式。这个项目将涉及许多数学领域，这个方向已经产生了这些领域独立感兴趣的结果，包括：麦克唐纳对称函数理论，热带组合学和惠特克函数，以及某些量子可积系统。自两百年前被发现以来，高斯分布(钟形曲线)已经成为数学、社会和科学的最大贡献之一。一个强有力的理论来解释和分析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性类。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更好地捕捉了从自然灾害到急诊室就诊的各种事件的随机性和严重性。最近，大量的研究工作集中在理解没有用任何经典发展的统计学来很好地描述的系统上。这些系统不符合经典描述的原因通常是自然可观测性与随机输入和噪声的潜在来源之间的非线性关系。关于复杂系统的各种模型，如生长过程、聚合物链、质量传输、交通流、排队论、驱动气体和湍流，在数学、物理、材料科学、化学和生物学中已经被活跃地研究了40多年。所有这些系统都不符合经典的高斯统计，因为已经从实验证据中观察到了湍流液晶、薄膜上的晶体生长、小平面边界、细菌菌落生长、纸张润湿、裂缝形成和燃烧前沿。令人惊讶的是，尽管它们不同，但所有的系统都属于一个新的统计普适性类别，其性质可以用一个称为Kardar-Parisi-Zhang(KPZ)方程的模型来描述。本项目的目的是发展对KPZ方程及其普适性类的统计理解。

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.