Recurrence and dynamical Borel-Cantelli results in dynamical systems

本项目旨在了解混沌动力系统的极值过程。范围在于数学分析和概率论的交界。在动力系统中，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相空间的特定区域，控制第一次返回该区域的次数的概率分布是什么？该项目旨在利用回归时间统计理论的最新发展来确定动力系统的（弱）收敛到一个极值过程。这种过程在极值理论中自然出现，并可用于确定由动力系统产生的最大值时间序列的概率性质。在适当归一化伯克霍夫和的情况下，产生的自然极限过程是布朗运动的极限过程。因此，本项目旨在了解时间序列的归一化最大值的相应极限过程。对于由独立同分布随机变量生成的时间序列，已知存在弱收敛至极值过程。然而，对于由动态系统产生的时间序列，需要新的思想来建立收敛性。该项目将发展一种理论来确定一般动力系统何时出现弱收敛到极值过程，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）和由常微分方程控制的连续时间混沌系统。该项目将使用动力系统的数学分析方法，对那些寻求在纯数学或应用数学方面工作的人，或那些在极端天气/气候的统计建模方面工作的人来说，这将是有益的。如果取得良好进展，该项目将探索低维天气模型中的极端情况（例如洛伦兹方程）。本项目采用动力系统和遍历理论中的理论方法。通过主管的联系将会有工业合作。这包括气象局和威利斯韬睿沃森通过当前的EPSRC项目（EP/P034489/1）：合作伙伴提供实物资金，以支持他们在项目上的时间，通过研讨会的参与，并通过定期的研究会议，发展理论的实际应用（例如天气/气候）。博士生将通过计划的研讨会和导师（EP/P034489/1上的PI）组织的会议参与这些组织。

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).