Random Dynamical Systems and Limit Theorems for Optimal Tracking

动力系统作为各种物理现象的重要数学模型，出现在诸如天气建模、系统生物学和统计物理学等领域。动态系统由状态空间和控制系统从一种状态向另一种状态演化的规则组成，状态空间中一个点代表系统状态的完整描述。该项目从两个互补的角度关注这些系统的长期行为。从第一个角度来看，该项目试图描述随机选择进化规则时典型系统的行为。这些结果揭示了人们期望在无序系统中发现的性质。第二种观点，“逆问题”，涉及从动态系统的观测中恢复一些信息的统计问题。虽然有许多动力系统被用作数学模型的例子，并且有大量关于推理和估计的统计文献，但统计程序在应用于非线性动力系统产生的数据时的性能却知之甚少。这个项目的重点是描述什么时候传统的统计程序可以有效地应用于动力系统。除了非常丰富的潜在应用之外，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项目的研究课题获得可靠的概率建模和统计推断方面的宝贵技能。符号动力系统可以根据可能轨迹上的离散约束来定义，作为随时间演变的系统的原型模型。随机选择约束条件可以产生这些系统的随机系综。离散概率和动力系统的思想可以用来分析高概率系统的结构特性。对于反问题，该项目旨在评估几种统计推理程序的性能，包括频率论和贝叶斯，当涉及的模型是动态系统。关于过程的收敛性和一致性的基本问题可以使用遍历理论中的工具来解决，例如连接和热力学形式论。

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.