Analysis of stochastic partial differential equations with multiple scales

在复杂系统的描述中，无论是确定性的还是随机的，通常重要的是能够有这些系统的简化模型，以使他们的分析更容易接近。通常，这样的简化是通过观察被认为与系统进化更相关的少数因素，并忽略其他被认为不太相关的因素来实现的。然而，这样的近似，在某个给定的时间间隔内是有效的，在更长的时间尺度上是无效的，并且被忽略的因素在系统行为的描述中起着基本的作用。本研究将分析这些模型中使用的一大类方程。这些都是非常复杂的方程，对它们的理解对于更深入地理解模型的主要特征和在应用中更好地有效至关重要。这项研究将发展涉及数学许多领域的新方法和新技术。教育和培训也将是该项目的主要部分。本研究项目的主要目的是分析多尺度随机偏微分方程的极限定理。特别地，PI将研究无限自由度系统的Smoluchowskii-Kramers近似的一些推广及其长期效应，以及对某些类别的随机偏微分方程（SPDEs）的平均和大偏差原理的有效性。具体来说，PI将试图了解在空间中噪声微弱且几乎为白色的状态下会发生什么。此外，她将研究在窄通道上定义的spde或在整个空间中描述随机不可压缩粘性流体的spde收敛到在图和开放书籍上定义的一类新的spde。这些渐近结果的重要性不仅在于为分子马达和流体动力学研究中出现的一些相关的多尺度spde提供简化描述，而且还在于在极限情况下，它们提供了值得研究的新的有趣的数学对象。这种解决渐近问题的方法的特点和统一之处在于努力理解它们是如何相互作用和相互作用的。

In the description of complex systems, both deterministic and stochastic, it is usually important to be able to have a simplified model of those systems, in order to make their analysis more approachable. Usually, such a simplification is realized by looking at a smaller number of factors that are considered more relevant for the evolution of the system and by neglecting other factors that are considered less relevant. However, such an approximation, that can be effective on some given time interval, is not effective on longer time scales and the neglected factors turn out to play a fundamental role in the description of the systems' behavior. This research will analyze a large class of equations used in these models. These are highly complex equations and an understanding of them is crucial for a deeper understanding of the main features of the model and for a better effectiveness in applications. This research will develop new methods and techniques ranging over many fields of mathematics. Education and training will also be a major part of the project. The main goal of this research project is the analysis of limit theorems for stochastic partial differential equations having multiple scales. In particular, the PI will study some generalizations of the Smoluchowskii-Kramers approximation for systems with an infinite number of degrees of freedom and its long-time effects, as well as the validity of the averaging and the large deviation principle for some classes of stochastic partial differential equations (SPDEs). Specifically, the PI will try to understand what happens in the regime where the noise is weak and almost white in space. Moreover, she will study the convergence of SPDEs defined on narrow channels or describing stochastic incompressible viscous fluids in the whole space to a new class of SPDEs defined on graphs and open books. These asymptotic results will be important not only to provide a simplified description of some relevant multi-scale SPDEs that arise e.g. in the study of molecular motors and fluid dynamics, but also because at the limit they provide new interesting mathematical objects that are worthy of investigation. What characterizes and unifies this approach to all of these asymptotic problems is the effort to understand how they all interplay and interact one with the other.