Multiscale Analysis of Infinite-Dimensional Stochastic Systems

无限维随机系统的多尺度分析

基本信息

批准号：
1954299
负责人：
Sandra Cerrai
金额：
$ 30万
依托单位：
University of Maryland, College Park
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954299&HistoricalAwards=false
关键词：
Multiscale Analysis Infinite Dimensional Stochastic

项目摘要

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.

在分析复杂系统时，重要的是要对其进行简化的描述。通常，通过考虑较小的因素被认为与对这些系统的理解更相关，并忽略所有被认为不相关的因素来实现这种简化。但是，某些因素可能在特定时间范围内看起来不太重要的因素，在更长的时间范围内起着至关重要的作用。因此，正确理解复杂系统的多个尺度之间的相互作用是至关重要的，以便具有更有效的模型。该项目将为随机部分微分方程介绍并开发新的渐近分析方法。这些都是高度复杂的对象，并且朝着简化和更有效的分析方向进行的任何努力对于他们的深入理解和广泛的应用程序范围都很重要。该分析需要开发新方法，并大量引入新技术，这些新技术必须在数学领域的许多领域中进行范围，从无限维空间的分析到随机分析到随机分析以及部分微分方程的理论。该项目为研究生提供了研究培训机会。该研究项目的主要目标是研究系统的几个渐近问题，这些问题由具有多个量表的随机部分微分方程描述。系统对给定时间尺度具有可忽略的影响的系统的小型随机性和确定性扰动可能在更长的时间范围内变得至关重要。限制了大偏差和标准化理论框架中的定理，平均原理的各种实现，小质量限制（如Smoluchowski-kramers近似值）将是研究的对象。特征和统一所有这些渐近问题的方法的特征和统一的是，努力了解它们如何相互作用和相互作用。该奖项反映了NSF的法定任务，并通过使用该基金会的知识分子的优点和更广泛的影响来审查标准，认为NSF的法定任务值得通过评估来获得支持。