Intermittency and Physical Properties of Stochastic Partial Differential Equations

随机偏微分方程的间歇性和物理性质

• 批准号: 1513556
• 负责人: Daniel Conus
• 金额: $ 13.9万
• 依托单位: Lehigh University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1513556&HistoricalAwards=false
• 关键词: Intermittency; Physical; Properties; Stochastic; Partial

This project studies physical properties of the solutions to stochastic partial differential equations (SPDEs), which have been a growing topic of research in probability over the last decades. These equations are partial differential equations involving a random component, known as the noise. They mathematically model dynamical systems in a wide spectrum of fields. Physics is the most important of these and was the origin of the use of SPDEs. These equations appear for instance in models for growth at interfaces (the Kardar-Parisi-Zhang equation), models for the movement of galaxies, and polymer models, and also in models of randomly forced propagation of waves, such as the movement of DNA in a fluid. SPDEs also appear in biology, for instance in predator-prey models and models for growth of bacterial populations. They are also a tool of importance in the modeling of interest rates in mathematical finance. This project focuses on study of the property of intermittency for solutions to SPDEs: the fact that the solution develops high-valued peaks (atypical of the average behavior) concentrated on small spatial regions. This phenomenon and its conjectured connection to turbulence and chaos has been very carefully described by physicists, who conjectured many results, only a very few of which are mathematically proved. This research project aims to obtain a more profound understanding of this phenomenon. This field being relatively young, some effort will also be invested into popularizing these ideas among the scientific community, in particular among students and future researchers. More specifically, this project will focus on equations related to the Kardar-Parisi-Zhang (KPZ) equation of physics, which is the central component of the KPZ universality class, a class of probabilistic models appearing in many instances of interface growth phenomena, such as percolation of a liquid in a porous medium, development of a population of bacteria, or the movement of cars in heavy traffic. In order to establish and understand the intermittency of solutions, several new techniques, such as stochastic Young inequalities, moment estimates, and scaling properties have been developed in recent years. These techniques have been successful in specific cases, but more general and more precise results are still sought. Knowledge of the quantitative behavior of the moments of the solution to the SPDE, as well as almost-sure behavior of this solution, are extremely important when it comes to careful study the peaking phenomenon, for instance the position, size, movement, and fractal dimension of the peaks. The project will develop new methods, such as the use of Galton-Watson type processes. The main objective is to carefully understand the impact of the noise on the physical properties of the solution. Thus, the project aims to compare the impact of different types of noise (e.g., white, colored, fractional) on properties of the associated equations.

该项目研究随机偏微分方程 (SPDE) 解的物理性质，在过去的几十年里，随机偏微分方程一直是概率研究中一个日益增长的话题。这些方程是涉及随机分量（称为噪声）的偏微分方程。他们对广泛领域的动力系统进行数学建模。物理学是其中最重要的，也是 SPDE 使用的起源。这些方程出现在例如界面生长模型（Kardar-Parisi-Zhang 方程）、星系运动模型和聚合物模型中，也出现在波的随机强制传播模型中，例如流体中 DNA 的运动。 SPDE 也出现在生物学中，例如捕食者-猎物模型和细菌种群生长模型。它们也是数学金融中利率建模的重要工具。该项目重点研究 SPDE 解的间歇性特性：该解产生集中在小空间区域的高值峰值（非典型的平均行为）。物理学家已经非常仔细地描述了这种现象及其与湍流和混沌的推测联系，他们推测了许多结果，但其中只有极少数得到了数学证明。本研究项目旨在更深入地了解这一现象。这个领域相对较年轻，我们还将投入一些努力在科学界，特别是学生和未来的研究人员中推广这些想法。更具体地说，该项目将重点关注与物理 Kardar-Parisi-Zhang (KPZ) 方程相关的方程，该方程是 KPZ 普适类的核心组成部分，这是一类出现在界面增长现象的许多实例中的概率模型，例如多孔介质中液体的渗透、细菌群体的发展或交通繁忙中的汽车运动。为了建立和理解解的间歇性，近年来开发了几种新技术，例如随机杨氏不等式、矩估计和缩放特性。这些技术在特定情况下取得了成功，但仍寻求更普遍和更精确的结果。当仔细研究峰化现象（例如峰的位置、大小、移动和分形维数）时，了解 SPDE 解矩的定量行为以及该解的几乎确定的行为极其重要。该项目将开发新方法，例如使用高尔顿-沃森型工艺。主要目标是仔细了解噪声对解决方案物理特性的影响。因此，该项目旨在比较不同类型的噪声（例如白色、彩色、分数）对相关方程属性的影响。