Asymptotic problems for stochastic partial differential equations

本研究项目的目标是开发新的方法来分析由偏微分方程描述的系统，在存在噪声（随机）扰动和小参数的情况下。我们感兴趣的是描述这类系统在参数消失时的不同行为，以及不同极限区域之间的相互作用。特别是，我们将研究对应用有重要意义的新数学问题，以及经典问题中的新效应。对于具有无限多个自由度的系统的这些问题的处理是一个相对较新的研究领域，它已经激起了许多领域的生动兴趣，也因为这些是非常复杂的对象，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在应用中更好的有效性都是重要的。我们的分析需要新方法的发展和新技术的大量引进，这些新技术必须涵盖数学的许多领域。我们的目标是研究由随机偏微分方程描述的一类广泛系统的小确定性和随机扰动。事实上，在一个时间尺度上可以忽略不计的小扰动，在更大的时间尺度上可能变得至关重要。我们以前的一些论文已经考虑了小扰动的长期影响，本项目必须被视为该计划的延续。极限定理，特别是大偏差理论，平均原理和它们之间的相互作用，以及Smoluchowskii-Kramers近似的几种推广是我们的主要工具。具有多/无限自由度的系统通常具有不同来源和不同阶次的扰动。这种受扰动系统的长期行为应该用近似层次来描述。另一方面，具有不稳定性的纯确定性系统在一定条件下的长期行为，应该用随机过程来描述。因此，该问题的自然通用性是考虑随机系统（不一定是确定性动力系统）的确定性和随机扰动。

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).