Recurrence and dynamical Borel-Cantelli results in dynamical systems

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).