Asymptotic problems for stochastic partial differential equations

随机偏微分方程的渐近问题

• 批准号: 0907295
• 负责人: Sandra Cerrai
• 金额: $ 25.5万
• 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: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0907295&HistoricalAwards=false
• 关键词: Asymptotic; problems; stochastic; partial; differential

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The objective of this research project is the investigation and the mathematical analysis of a wide class of asymptotic problems for stochastic partial differential equations, with emphasis on the study of averaging principle, large deviation principle, stochastic singular perturbation and the interplay between them. Multi-scale and asymptotic methods in the analysis of deterministic and random systems are getting more and more important and by now they are acknowledged among the most powerful tools in many fields. They play an important role in the study of systems which possess multiple scales, in the fundamental effort to give a simplified description of them. With this project the PI is trying to apply multi-scale methods to the analysis of a large class of SPDE's: these are highly complex mathematical objects and any effort which goes in the direction of their simplification is crucial for a deeper understanding of the main features of the models and for a better effectiveness in applications. Large deviations, averaging, homogenization, singular perturbations and in general multi-scaling limits have been widely studied for systems with a finite number of degrees of freedom and a wide literature on this subject is available. The treatment of the analogous problems in the infinite dimensional case (with maybe the only exception of large deviations) is a relatively new field of investigation, which has already stirred up a vivid interest in many fields. Due to the ampleness of the subject, which is related to many branches of mathematics, and the diversified range of their applications in physics, biology and engineering, these researches are only at the beginning and a lot of work has still to be done. The analysis requires the development of new methods and the substantial introduction of new techniques which have to range over many fields in mathematics, from stochastic analysis, to the theory of partial differential equations, to analysis in infinite dimensional spaces.

该奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。本研究项目的目的是调查和数学分析一大类随机偏微分方程渐近问题，重点研究平均原理、大偏差原理、随机奇异摄动以及它们之间的相互作用。多尺度和渐近方法在确定性和随机系统的分析中变得越来越重要，到目前为止，它们已经成为许多领域公认的最强大的工具之一。它们在研究具有多个尺度的系统中起着重要的作用，这是对它们进行简化描述的根本努力。通过这个项目，PI试图将多尺度方法应用于对一大类SPDE的分析：这些是高度复杂的数学对象，任何朝着简化方向进行的努力对于更深入地理解模型的主要特征和更好地应用效果都是至关重要的。对于有限个自由度系统的大偏差、平均化、齐次化、奇异摄动和一般的多尺度极限已被广泛地研究，并且有大量关于这一主题的文献。无限维类问题的处理(大偏差可能是唯一的例外)是一个相对较新的研究领域，已经在许多领域引起了人们的浓厚兴趣。由于该学科涉及面广，涉及到数学的多个分支，在物理、生物、工程等领域的应用也很广泛，这些研究还处于起步阶段，还有很多工作要做。这种分析需要新方法的发展和新技术的大量引入，这些新技术必须涵盖数学的许多领域，从随机分析到偏微分方程组理论，再到无限维空间的分析。