Statistical Properties of Dynamical Systems

这个项目的目标是了解确定性系统中随机性的起源，特别注意可用于模拟物理和生物系统中的传输现象的扩展系统，以及其演化规律随时间变化的系统。首席研究人员将使用概率和几何学的工具来开发预测复杂系统统计行为的方法。该项目为研究生提供培训，大部分研究将与研究生、博士后研究员和其他职业生涯早期的数学家合作完成。该项目由四个部分组成：(I)中心极限定理的收敛速度；(Ii)保持无限测量的系统的动力学；(Iii)统计性质的灵活性；以及(Iv)随机介质的传输。该项目的统一主题是动力学和概率之间的相互作用。概率论方法，如精化极限定理和耦合，被用于研究确定性系统，而动力学方法，包括转移算子的谱性质和格子空间上的重整化动力学，被用来处理概率问题。在第(I)和(Ii)部分中获得的成果将应用于该项目的第(Iii)和(Iv)部分。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。

The goal of this project is to understand the origin of stochasticity in deterministic systems, with particular attention to extended systems that can be used to model transport phenomena in physical and biological systems and to systems whose evolution laws change with time. The principal investigator will use tools from probability and geometry in order to develop methods to predict the statistical behavior of complicated systems. The project provides training for graduate students, and a significant portion of the research will be done in collaboration with graduate students, postdoctoral fellows, and other early-career mathematicians.The project consists of four parts:(I) The rate of convergence in the Central Limit Theorem;(II) Dynamics of systems preserving infinite measure;(III) Flexibility of statistical properties; and(IV) Transport of random media.The unifying theme of the project is the interplay between dynamics and probability. Probabilistic techniques, such as refined limit theorems and coupling, are used to the study of deterministic systems, and dynamical methods including spectral properties of transfer operators and renormalization dynamics on the space of lattices are used to handle probabilistic questions. Results obtained in parts (I) and (II) will be applied in parts (III) and (IV) of the project.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.