Multiscale Analysis of Infinite-Dimensional Stochastic Systems

在分析复杂系统时，重要的是能够对它们进行简化描述。通常，这种简化是通过考虑被认为与理解这些系统更相关的较少数量的因素以及忽略所有被认为不太相关的因素来实现的。然而，有些因素在一定的时间尺度上可能显得不那么重要，但在更长的时间尺度上却起着至关重要的作用。因此，正确理解复杂系统多尺度间的相互作用是建立更有效模型的基础。本计画将介绍及发展随机偏微分方程式渐近分析的新方法。这些都是高度复杂的对象，任何朝着简化和更有效的分析方向的努力都是重要的，无论是对它们的深入理解还是对它们可能的广泛应用。这种分析需要发展新的方法和大量引进新的技术，这些技术必须涉及数学的许多领域，从无限维空间的分析到随机分析和偏微分方程理论。该项目为研究生提供研究培训机会。本研究计画的主要目的是研究多尺度随机偏微分方程所描述系统的几个渐近问题。系统的小的随机和确定性扰动，在给定的时间尺度上可以忽略不计的影响，可以成为一个更长的时间尺度上的关键。 极限定理的大偏差和亚稳性理论的框架内，各种实现的平均原则，小质量的限制，在Smoluchowski-Kramers近似将是研究的对象。什么特点和统一的所有这些渐近问题的方法是努力了解它们如何相互作用和相互影响的。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

When analyzing complex systems, it is important to be able to have a simplified description of them. Oftentimes, such a simplification is realized by considering a smaller number of factors that are considered more relevant to the understanding of these systems and by neglecting all those factors that are considered less relevant. However, some factors that may appear less important at a certain time scale, turn out to play a crucial role at some longer time scale. Thus, it is fundamental to understand correctly the interplay among multiple scales of complex systems in order to have more effective models. This project will introduce and develop new methods of asymptotic analysis for stochastic partial differential equations. These are highly complex objects and any effort that goes in the direction of their simplified and more effective analysis is important, both for their deeper understanding and for the wide range of their possible applications. This analysis requires the development of new methods and the substantial introduction of new techniques which have to range over many fields in mathematics, from analysis in infinite-dimensional spaces to stochastic analysis and the theory of partial differential equations. The project provides research training opportunities for graduate students. The main goal of this research project is the study of several asymptotic problems for systems that are described by stochastic partial differential equations having multiple scales. Small stochastic and deterministic perturbations of a system, which have a negligible effect on a given time scale can become crucial on a longer time scale. Limit theorems in the framework of the theory of large deviation and metastability, various realizations of the averaging principle, small mass limits as in the Smoluchowski-Kramers approximation will be the objects of the research. What characterizes and unifies the present approach to all these asymptotic problems is the effort to understand how they all interplay and interact with each other.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.