Intermittency and Physical Properties of Stochastic Partial Differential Equations

本计画主要研究随机偏微分方程解的物理性质，这是近十年来机率论研究的热门课题。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，聚合物模型，以及波的随机强制传播模型中，例如DNA在流体中的运动。SPDE也出现在生物学中，例如捕食者-猎物模型和细菌种群增长模型。它们也是数学金融中利率建模的重要工具。该项目的重点是研究SPDE解决方案的不稳定性的性质：解决方案开发高值峰（平均行为的非典型性）集中在小空间区域的事实。这种现象及其与湍流和混沌的联系已经被物理学家非常仔细地描述过，他们提出了许多结果，其中只有很少一部分得到了数学证明。本研究项目旨在更深入地了解这一现象。这一领域相对较新，还将作出一些努力，在科学界，特别是在学生和未来的研究人员中推广这些想法。更具体地说，该项目将侧重于与Kardar-Parisi-Zhang（KPZ）物理方程相关的方程，该方程是KPZ普适性类的核心组成部分，这是一类出现在许多界面生长现象实例中的概率模型，例如液体在多孔介质中的渗透，细菌种群的发展或汽车在繁忙交通中的运动。为了建立和理解解的非线性，近年来发展了一些新的技术，如随机Young不等式，矩估计和标度性质。这些技术在特定情况下是成功的，但仍在寻求更普遍和更精确的结果。了解SPDE解的矩的定量行为，以及该解的几乎确定的行为，对于仔细研究峰化现象（例如峰的位置、大小、运动和分形维数）是极其重要的。该项目将开发新的方法，例如使用Galton-Watson型过程。主要目的是仔细了解噪声对溶液物理性质的影响。因此，该项目旨在比较不同类型的噪音（例如，白色、彩色、分数）对相关方程的性质的影响。

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.