Analysis of stochastic partial differential equations with multiple scales

在描述复杂系统时，无论是确定性的还是随机性的，通常重要的是能够有这些系统的简化模型，以便使它们的分析更接近。通常，这种简化是通过查看被认为与系统演化更相关的较少数量的因素，并忽略被认为不太相关的其他因素来实现的。然而，这样的近似，可以有效的一些给定的时间间隔，是不是有效的更长的时间尺度和被忽视的因素，在描述系统的行为中发挥了根本性的作用。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，理解它们对于更深入地理解模型的主要特征和提高应用效率至关重要。这项研究将发展新的方法和技术，涉及许多数学领域。教育和培训也将是该项目的一个主要部分。本研究计画的主要目标是分析多重尺度随机偏微分方程的极限定理。特别是，PI将研究Smoluchowskiii-Kramers近似的一些推广，用于具有无限多个自由度的系统及其长期影响，以及某些随机偏微分方程（SPDE）的平均和大偏差原理的有效性。具体来说，PI将试图了解在噪声较弱且几乎为空间白色的情况下会发生什么。此外，她将研究收敛的SPDE定义在狭窄的通道或描述随机不可压缩粘性流体在整个空间中的一类新的SPDE定义的图形和开放的书籍。这些渐近结果将是重要的，不仅提供了一些相关的多尺度SPDE，例如，在分子马达和流体动力学的研究中出现的简化描述，但也因为在极限，他们提供了新的有趣的数学对象，是值得调查。对所有这些渐近问题的这种方法的特点和统一是努力理解它们如何相互作用和相互作用。

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.