Recurrence and dynamical Borel-Cantelli results in dynamical systems

这个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的接口。动力系统中一个具有挑战性和活跃的领域是返回时间统计。也就是说，给定一个动力学系统和一个特定的相空间区域，控制第一次返回到这个区域的时间的概率分布是什么？该项目的目的是利用最近的发展理论的返回时间统计，以确定（弱）收敛到一个极值过程的动力系统。这样的过程在极值理论中自然出现，并且可以用于确定由动力系统生成的最大值的时间序列的概率特性。在适当规范化的伯克霍夫和的情况下，自然极限过程是布朗运动。因此，该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，已知会发生弱收敛到极值过程。然而，对于由动力系统生成的时间序列，需要新的想法来建立收敛。本计画将发展一个理论，以决定一般动力系统何时会弱收敛至极值过程，并将此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，并将有利于那些寻求在纯数学或应用数学工作，或那些在极端天气/气候的统计建模工作。如果进展良好，该项目将探索低维天气模型（例如Lorenz方程）中的极端情况。该项目使用动力系统和遍历理论中的理论方法。将通过主管的联系进行工业合作。这包括英国气象局和Willis Towers沃森通过当前的EPSRC项目（EP/P034489/1）：合作伙伴通过参加研讨会和定期研究会议来支持他们在项目上的时间（例如天气/气候）。博士生将通过计划中的研讨会和由主管（EP/P034489/1上的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).