Asymptotic problems for stochastic partial differential equations

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。本课题的研究目的是对一类广泛的随机偏微分方程的渐近问题进行研究和数学分析，重点研究了平均原理、大偏差原理、随机奇异摄动及其相互作用。多尺度和渐近方法在确定性和随机系统的分析中越来越重要，目前已被公认为是许多领域中最强大的工具之一。它们在具有多尺度的系统的研究中起着重要的作用，从根本上努力给出它们的简化描述。在这个项目中，PI试图将多尺度方法应用于分析一大类SPDE：这些是高度复杂的数学对象，任何朝着简化方向的努力对于更深入地理解模型的主要特征和更好的应用效率至关重要。对于具有有限自由度的系统，大偏差、平均、均匀化、奇异摄动和一般的多尺度极限已经得到了广泛的研究，并且有大量关于这一主题的文献。在无限维情况下处理类似问题（可能只有大偏差例外）是一个相对较新的研究领域，它已经在许多领域激起了浓厚的兴趣。由于该学科涉及面广，涉及数学的多个分支，在物理、生物、工程等领域的应用也比较广泛，所以这些研究才刚刚起步，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进，这些技术必须涵盖数学的许多领域，从随机分析到偏微分方程理论，再到无限维空间的分析。

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.