Recurrence and dynamical Borel-Cantelli results in dynamical systems

这个项目的目的是了解混沌动力系统的极值过程。它的范围在于数学分析和概率的交界处。在动态系统中，返回时间统计是一个具有挑战性和活跃的领域。也就是说，给定一个动力系统和一个特定的相空间区域，控制第一次返回该区域的时间的概率分布是什么？该项目旨在利用返回时间统计理论的最新发展来确定动力系统到极值过程的(弱)收敛。这种过程是在极值理论中自然产生的，可以用来确定由动力系统产生的极大值时间序列的概率性质。在适当归一化Birkhoff和的情况下，产生的自然极限过程是布朗运动的过程。因此，本课题旨在了解时间序列的归一化极大值对应的极限过程。对于由独立同分布随机变量产生的时间序列，已知会发生弱收敛到极值过程。然而，对于由动力系统产生的时间序列，需要新的思想来建立收敛。这个项目将开发一种理论来确定一般动力系统何时发生弱收敛到极值过程，并将该理论应用于特定的例子，如离散时间双曲动力系统(例如Anosov和Axiom A系统)，以及由常微分方程式控制的连续时间混沌系统。本项目将使用动力系统的数学分析方法，对那些寻求从事纯数学或应用数学工作的人，或那些从事天气/气候极端统计建模工作的人有益。如果进展顺利，该项目将探索低维天气模型(例如洛伦兹方程)中的极端情况。该项目使用动力系统中的理论方法和遍历理论。将通过牵头主管的联系人进行行业协作。这包括英国气象局和Willis Towers Watson通过目前的EPSRC项目(EP/P034489/1)：在该项目中，合作伙伴通过参加研讨会和定期召开研究会议，开发理论的实际应用(例如，对天气/气候)，提供实物资金，支持他们在该项目上的时间。这位博士生将通过计划中的研讨会和主管(EP/P034489/1的PI)组织的会议与这些组织接触。

This project aims to understand extremal processes for chaotic dynamical systems. The scope lies at the interface of mathematical analysis and probability. A challenging and active area in dynamical systems is that of return time statistics. Namely, given a dynamical system and a specific region of phase space, what is the probability distribution that governs the times of first return to this region? The project aims to exploit recent developments on the theory of return time statistics to determine (weak) convergence to an extremal process for dynamical systems. Such processes arise naturally within extreme value theory, and can be used to determine the probabilistic properties of the time series of maxima, as generated by the dynamical system. In the case of suitably normalised Birkhoff sums the natural limit process arising is that of a Brownian motion. Thus this project aims to understand the corresponding limit processes for normalised maxima of the time series. For time series generated by independent identically distributed random variables, weak convergence to an extremal process is known to occur. However, for time series generated by a dynamical system new ideas are needed to establish convergence. This project will develop a theory to determine when weak convergence to an extremal process occurs for a general dynamical system, and apply the theory to particular examples such as discrete time hyperbolic dynamical systems (e.g. Anosov and Axiom A systems), and continuous time chaotic systems governed by ordinary differential equations.This project will use approaches in mathematical analysis of dynamical systems, and will be of benefit to those seeking to work in pure or applied mathematics, or to those working within statistical modelling of extremes for weather/climate. Given good progress, the project will explore extremes in low dimensional weather models (e.g. Lorenz equations).This project uses theoretical approaches within dynamical systems and ergodic theory. There will be industrial collaboration via the contacts of the lead supervisor. This includes the Met Office, and Willis Towers Watson through the current EPSRC project (EP/P034489/1): where the partners have provided in-kind funding to support their time on the project through workshop participation, and through regular research meetings on the development of practical applications of the theory (e.g. to weather/climate). The PhD student will engage with these organisations through planned workshops, and meetings organised by the supervisor (who is PI on EP/P034489/1).