Qualitative Properties of Stochastic Partial Differential Equations

本文研究了随机偏微分方程（SPDE）中的三个问题。 第一个问题涉及独特性。 除非SPDE有独特的解决方案，否则它们对建模毫无用处。 最近的工作由提议者，L。Mytnik和E.珀金斯解决了一个长期存在的问题，即与热方程有关的SPDE的唯一性。 使用这些新开发的工具，提议者将攻击其他类型的方程的唯一性问题，并研究非负解之间的唯一性。 第二个问题涉及行波，它广泛存在于物理系统中。 最近，该提议者L. Mytnik和J. Quastel解决了Brunet和Derrida提出的一个问题，涉及噪声项小时行波的渐近速度。 提议者将希望将这种分析扩展到其他方程和相关的粒子系统。 第三个主题涉及随机波动方程。 最常被研究的SPDE是热方程的变体，但其他如随机波动方程正受到越来越多的关注。 与D. Geba，提议者将喜欢研究涉及零形式的随机波动方程。 利用Bourgain引入的一些函数空间，提出者的目标是研究短时存在性。 所发展的思想应该与其他类型的方程有关。 SPDE的领域正在迅速扩大。 随着技术向微观水平发展，随机噪声的影响变得越来越重要。 由于我们所拥有的最重要的数学建模工具是常微分方程和偏微分方程，因此SPDE的研究在许多应用领域中变得必不可少。 尽管SPDE已经有几十年的历史，但该地区直到最近才受到广泛关注。 仍然需要开展开拓性工作，为该领域建立一个工具箱。 该建议涉及SPDE中的三种类型的问题。 提出者认为，通过研究这些具体的例子是正确的方法，以产生方法和进一步我们的理解，无论是理论和应用。 总之，提出者认为，SPDE将在未来的数学应用中发挥重要作用。 他希望发展这一理论，并帮助培养研究生，使这一领域能够发挥其潜力。

This proposal deals with three topics in stochastic partial differential equations (SPDE). The first question deals with uniqueness. Unless SPDE have unique solutions, they are useless for modeling. Recent work by the proposer, L. Mytnik, and E. Perkins settled a longstanding question about uniqueness of SPDE related to the heat equation. Using these newly developed tools, the proposer will attack uniqueness questions for other types of equations, and study uniqueness among nonnegative solutions. The second question deals with traveling waves, which occur widely in physical systems. Recently the proposer, L. Mytnik, and J. Quastel have solved a problem raised by Brunet and Derrida, involving the asymptotic speed of traveling waves when the noise term is small. The proposer will would like to extend this analysis to other equations, and related particle systems. The third topic involves stochastic wave equations. The most commonly studied SPDE are variants of the heat equation, but others such as the stochastic wave equation are receiving increasing attention. With D. Geba, the proposer will like to study stochastic wave equations involving null forms. Using some function spaces introduced by Bourgain, the proposer's goal is to study short-time existence. The ideas developed should be relevant to other classes of equations. The field of SPDE is rapidly expanding. As technology moves towards the micro level, the effect of random noise becomes more and more important. Since the most important mathematical modeling tools we have are ordinary and partial differential equations, the study of SPDE is becoming essential in many applied fields. Even though SPDE is now several decades old, the area has only recently received widespread attention. There is still a need for pioneering work to establish a toolbox for the area. This proposal deals with three types of problems in SPDE. The proposer believes that through the study of such specific examples is the right way to generate methods and further our understanding, for both theory and applications. In summary, the proposer believes that SPDE will play an essential role in future applications of mathematics. He wishes to develop the theory and help to train graduate students so that this area can fulfill its potential.