Non-Stationary Random Dynamical Systems and Applications

随机性和不可预测性因素始终存在，无法完全避免。因此，通过应用不同的地图，即随时间变化的变换，而不是重复应用完全相同的变换，可以更准确地解释许多现实生活中系统的发展。使用典型的映射序列来检查这些问题是直观的，这将我们带入随机动力学的概念。随机动力系统的理论可以追溯到乌兰和冯·诺伊曼，以及卡库塔尼。从那时起，它变成了一个发展良好的数学领域，有着无数的应用。但是，如果随机因素本身的分布和强度并不是恒定的，而是随着时间的变化而变化的，那该怎么办呢？这可以用非平稳随机动力系统来建模，研究这类系统的性质及其应用是本课题的主要目标。指导学生将是该项目的重要组成部分。许多与该项目密切相关的问题将被建议给研究生和本科生，以开始和增加他们在数学研究中的参与。在该项目的第一部分，目标是证明一个光滑的随机动力系统，在没有不变度量的假设下，肯定会导致随机合成的范数几乎肯定地指数增长，即使在非平稳的情况下也是如此。此外，还将研究平稳测度的连续模。项目的第二部分将导致关于随机矩阵乘积的非平稳版本的Furstenberg定理，该定理声称存在一个非随机序列，即Lyapunov指数的非平稳模拟，几乎肯定地描述随机矩阵乘积的范数的行为。中心极限定理、重对数律和其他极限定理的非对易版本将在非平稳设置下被证明，对于随机矩阵乘积，更一般地，对于李群上的随机游动。项目的第三部分将为非平稳的Anderson模型(包括Anderson-Bernoulli情况)提供频谱和动力学局部化的证明。此外，还讨论了非平稳Anderson模型的谱和本质谱的拓扑结构问题。特别是，它将表明，对于具有由独立同分布随机变量序列加上准周期背景给出的势的离散薛定谔算子，频谱必须由有限数量的区间组成，而对于周期背景，它可以有无限多个区间。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Random and unpredictable factors are always present and cannot be completely avoided. Therefore, the development of numerous real-life systems can be more accurately explained through the application of distinct maps, that is, of transformations that vary with time, rather than by repeatedly applying the exact same transformation. It is intuitive to examine these issues using typical sequences of maps, which brings us to the concept of random dynamics. Theory of random dynamical systems goes back to Ulam and von Neumann, as well as to Kakutani. Since then, it turned into a well-developed area of mathematics that has numerous applications. But what if the distribution and strength of the random factors itself does not remain constant, but changes with time? That can be modeled by non-stationary random dynamical systems, and the study of the properties of such systems and of their applications is the main goal of this project. Mentoring students will be an essential part of the project. Numerous questions closely related to the proposed project will be suggested to graduate and undergraduate students initiating and increasing their involvement in research in mathematics.In the first part of the project the goal will be to show that a smooth random dynamical system, under an assumption of absence of invariant measures, must lead to almost sure exponential growth of the norms of random compositions, even in non-stationary cases. Also, moduli of continuity of stationary measures will be studied. The second part of the project will lead to a non-stationary version of the Furstenberg Theorem on random matrix products, that claims that there exists a nonrandom sequence, a non-stationary analog of the Lyapunov exponent, that almost surely describes the behavior of the norms of random matrix products. Non-commutative versions of the Central Limit Theorem, the Iterated Logarithm Law, and other limit theorems will be proven in non-stationary settings, for random matrix products, and, more generally, for random walks on Lie groups. The third part of the project will provide the proof of Spectral and Dynamical Localization for the non-stationary Anderson Model (including the Anderson-Bernoulli case). Besides, the questions on topological structure of the spectrum and essential spectrum of the non-stationary Anderson Model will be addressed. In particular, it will be shown that for a discrete Schrödinger operator with potential given by a sequence of independent identically distributed random variables plus a quasiperiodic background the spectrum must consist of a finite number of intervals, while for a periodic background it can have infinitely many gaps.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.