Random Dynamical Systems and Limit Theorems for Optimal Tracking

动力系统是天气模拟、系统生物学和统计物理等领域中各种物理现象的重要数学模型。动力系统由状态空间和控制系统从一种状态演化到另一种状态的规则组成，在状态空间中的一个点表示系统状态的完整描述。本项目从两个互补的角度关注此类系统的长期行为。从第一个角度来看，该项目寻求描述随机选择进化规则时典型系统的行为。这样的结果阐明了人们可能会在无序系统中找到什么性质。第二种观点是“逆问题”，它涉及从动态系统的观测中恢复某些信息的统计问题。虽然有许多动力系统被用作数学模型的例子，并且有大量关于推断和估计的统计文献，但当统计过程应用于由非线性动力系统产生的数据时，人们对其性能了解甚少。这个项目的重点是确定传统统计程序何时可以有效地应用于动力系统。除了非常丰富的潜在应用，该项目还将对研究生的培训产生更广泛的影响，这些研究生将通过研究该项目的研究主题获得可靠的概率建模和统计推理方面的宝贵技能。符号动力系统可以根据对可能轨迹的离散约束来定义，它是随着时间演变的系统的原型模型。可以通过随机选择约束来产生这些系统的随机系综。然后，来自离散概率和动力系统的思想可以用来以高概率分析所得到的系统的结构属性。对于反问题，这个项目试图评估几种统计推理过程的性能，包括频率统计和贝叶斯统计，当涉及的模型是动态系统时。关于程序的收敛和一致性的基本问题可以使用遍历理论的工具来解决，例如连接和热力学形式主义。

Dynamical systems serve as important mathematical models for a wide variety of physical phenomena, arising in such areas as weather modeling, systems biology, and statistical physics. A dynamical system consists of a state space, in which a point represents a complete description of the state of the system, and a rule governing the evolution of the system from one state to another. This project focuses on the long-term behavior of such systems from two complementary points of view. From the first point of view, the project seeks to describe the behavior of typical systems when the rules of evolution are chosen at random. Such results shed light on what properties one might expect to find in disordered systems. The second point of view, the "inverse problem," concerns the statistical problem of recovering some information from the observation of a dynamical system. While there are many examples of dynamical systems being used as mathematical models, and there is a large statistical literature regarding inference and estimation, the performance of statistical procedures when applied to data generated by nonlinear dynamical systems is poorly understood. This project focuses on characterizing when traditional statistical procedures may be effectively applied in the context of dynamical systems. Beyond the very fertile potential applications, the project will also have broader impact on training of graduate students who will acquire invaluable skills in sound probabilistic modeling and statistical inference by working on the project's research topics.Symbolic dynamical systems, which may be defined in terms of discrete constraints on the possible trajectories, serve as prototypical models of systems that evolve over time. Random ensembles of these systems may be produced by selecting the constraints at random. Ideas from both discrete probability and dynamical systems may then be used to analyze the structural properties of the resulting systems with high probability. For the inverse problem, this project seeks to evaluate the performance of several statistical inference procedures, both frequentist and Bayesian, when the models involved are dynamical systems. Fundamental questions about convergence and consistency of the procedures may be addressed using tools from ergodic theory, such as joinings and the thermodynamic formalism.