Topics in Finite and Infinite Dimensional Random Dynamical Systems

随机动力系统出现在物理学、生物学、经济学等许多现象的建模中，当考虑到不确定性或随机影响（称为噪声）时。这些随机效应的引入不仅是为了弥补某些确定性模型的缺陷，而且往往是相当内在的现象。研究随机动力系统轨道的动力学行为是一个基本问题。这个项目的目的是建立一些基本的几何框架的无限维随机动力系统。通过证明随机不变流形和叶理的存在性和鲁棒性，得到确定性系统在随机扰动下的结构稳定性的结果。包含乘法遍历定理的Floquet理论将被开发，随机吸引子的统计特性，以及随机介质中的图案形成将被探索。此外，还将发展有限维随机动力系统的光滑共轭理论，该项目的更广泛影响包括在这一新兴动力学领域培训研究生和初级教员。上述许多领域的应用将创造机会，在许多层面上与来自其他学科的学生和教师进行互动，并将导致技术进步。一个这样的例子是纳米器件研究的可能应用，例如，为了准确地预测性能，必须考虑随机热波动或量子效应。这一领域是许多领域之一，完全可以使用这里将要开发的这类工具进行分析。

Random dynamical systems arise in the modeling of many phenomena in physics, biology, economics, etc. when uncertainties or random influences, called noises, are taken intoaccount. These random effects are not only introduced to compensate for the defects in some deterministic models, but also are often rather intrinsic phenomena. A fundamental problem is to study the dynamical behavior of orbits of random dynamical systems. This project seeks to establish some of the basic geometric framework for infinite dimensional random dynamical systems. By proving existence and robustness of random invariant manifolds and foliations, results on structural stability of deterministic systems under random perturbations will be sought. Floquet theory containing the multiplicative ergodic theorem will be developed, statistical properties of random attractors, and pattern formation in random media will be explored. The theory of smooth conjugacy for finite dimensional random dynamical systems will also be developed.The broader impacts of this project include the training of graduate students and junior faculty members in this emerging area of dynamics. Applications to the many fields mentioned above will create opportunities to interact on many levels with students and faculty members from other disciplines and will lead to advances in technology. One such example are the possible applications to the study of nano-devices, for instance, random thermal fluctuations or quantum effects must be accounted for in order to accurately predict performance. That field is one of many that is completely open to analysis using tools of the type to be developed here.