Asymptotic problems for stochastic partial differential equations

随机偏微分方程的渐近问题

• 批准号: 1407615
• 负责人: Sandra Cerrai
• 金额: $ 32.1万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1407615&HistoricalAwards=false
• 关键词: Asymptotic; problems; stochastic; partial; differential

The objective of this research project is the development of new methods for the analysis of systems described by partial differential equations, in the presence of a noisy (random) perturbation and small parameters. We are interested in the description of the different behaviors of such systems, when the parameters are vanishing, and of the interplay between different limiting regimes. In particular, we will study new mathematical problems which are important for applications, as well as new effects in classical problems. The treatment of these problems for systems with an infinite number of degrees of freedom is a relatively new field of investigation, which has already stirred up a vivid interest in many fields, also because these are very complex objects and any effort that goes in the direction of their simplification is important for a deeper understanding of the main features of the models and for a better effectiveness in applications. Our analysis requires the development of new methods and the substantial introduction of new techniques which have to range over many fields in mathematics. Our goal in this proposal is studying small deterministic and stochastic perturbations of a wide class of systems described by stochastic partial differential equations. As a matter of fact, small perturbations, which are negligible on one time scale, can become crucial on a larger time scale. The long-time influence of small perturbations has been considered in a number of our previous papers, and the present project has to be considered as a continuation of this program. Limit theorems, especially the large deviation theory, the averaging principle and the interplay between them, as well as several generalizations of the Smoluchowskii-Kramers approximation are our main tools. Systems with many/infinite degrees of freedom often have perturbations of different origin and different order. Long-time behavior of such perturbed systems should be described by a hierarchy of approximations. On the other hand, long-time behavior of pure deterministic systems with instabilities, under certain conditions, should be described by a stochastic process. Therefore, the natural generality for the problem is in considering both deterministic and stochastic perturbations of stochastic systems (not necessarily deterministic dynamical systems).

该研究项目的目的是开发新方法，用于在存在嘈杂（随机）扰动和小参数的情况下通过部分微分方程描述的系统。当参数消失以及不同限制方案之间的相互作用时，我们对此类系统的不同行为的描述感兴趣。特别是，我们将研究对应用至关重要的新数学问题，以及对经典问题的新效果。对于具有无限数量的自由度的系统，对这些问题的处理是一个相对较新的调查领域，它已经激起了许多领域的生动兴趣，也是因为这些物体是非常复杂的对象，并且在简化方向上付出的任何努力对于对模型的主要特征和应用程序的有效性的更深入了解至关重要。我们的分析需要开发新方法，并大量引入新技术，这些技术必须在数学领域的许多领域中进行范围。我们在该提案中的目标是研究由随机部分微分方程描述的广泛的系统的小型确定性和随机扰动。实际上，一个时间尺度可以忽略的小扰动在更大的时间范围内变得至关重要。在我们以前的许多论文中已经考虑了小型扰动的长期影响，并且必须将目前的项目视为该计划的延续。限制定理，尤其是大偏差理论，平均原理和它们之间的相互作用，以及Smoluchowskii-Kramers近似的几种概括是我们的主要工具。具有许多/无限程度的自由度的系统通常具有不同的起源和不同顺序的扰动。这种干扰系统的长期行为应通过近似层次结构来描述。另一方面，在某些条件下，具有不稳定性的纯确定性系统的长期行为应通过随机过程来描述。因此，该问题的自然普遍性在于考虑随机系统的确定性和随机扰动（不一定是确定性动力学系统）。