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

kardar - paris - zhang （KPZ）方程是一个非线性随机偏微分方程，其统计性质被认为可以描述一大类数学模型，包括相互作用粒子系统、随机生长模型、定向聚合物和分支扩散过程。该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，当以各种重要类型的初始数据开始时，该理论应该得出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对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。自从两百年前高斯分布（钟形曲线）被发现以来，它就代表了数学对社会和科学最伟大的贡献之一。一个强有力的理论，解释和分析了世界上固有的随机性。用高斯统计准确描述的物理和数学系统被称为高斯普适性类。然而，本课程并不是包罗万象的。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就诊等事件的随机性和严重性。最近，重要的研究工作集中在理解那些不能很好地用任何经典统计学描述的系统上。这些系统之所以不符合经典描述，通常是由于自然观测值与随机输入和噪声的潜在来源之间存在非线性关系。四十多年来，数学、物理、材料科学、化学和生物学领域对复杂系统的各种模型进行了积极的研究，如生长过程、聚合物链、质量输运、交通流、排队理论、驱动气体和湍流等。所有这些系统都不符合经典的高斯统计，正如通过实验证据所观察到的那样，包括湍流液晶、薄膜上的晶体生长、facet边界、细菌菌落生长、纸张润湿、裂纹形成和燃烧锋。令人惊讶的是，尽管存在差异，所有的系统都属于一个新的统计普世性类，其性质用一个称为kardar - paris - zhang （KPZ）方程的单一模型来描述。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.