Dynamical Multivariate Extremes

The study of extreme events is of crucial importance in a multitude of scenarios where their occurrence has a serious disruptive effect. This is the case of natural hazards such as earthquakes, storms, draughts, pandemics or human-made disasters such as industrial and transport accidents, oil spills, nuclear explosions, financial crashes, etc.

These phenomena correspond to very peculiar states of systems whose time evolution is, often, accurately described by mathematical models called dynamical systems. Namely, whenever the orbits of the system (the several successions of states through which the system goes during a certain realisation) hit small critical regions of the phase space corresponding to abnormal configurations, one observes extreme or rare events. The critical regions are neighbourhoods of a critical set, which we will denote by M corresponding to configurations where appropriate observable functions achieve their maximum or minimum, i.e., their extremes. These sensitive regions have small measure, which also justifies the use of the word rare.

The study of rare events for dynamical systems is recent but has experienced a vast development in the last decade and motivated, in particular, applications to climate dynamics. This development has been anchored in a connection between the observation of rare events, detected by the appearance of extreme values, and the recurrence properties of the like sensitive regions, when submitted to the action of the underlying dynamics. The main idea is that chaotic systems lose memory quickly which makes their orbits behave as random asymptotically independent observations and then one can recover the classical results from Extreme Value Theory. This strategy has been successfully applied to prove the existence of limit theorems on the distributional limit of the extremal order statistics, point processes counting the number of rare events, and ergodic averages of heavy tailed observables. In some sense, the study of extreme events for dynamical systems has recently caught up with the state of the art of univariate Extreme Value Theory. However, since the 1980s extreme value theorists have gone in various directions: on the more theoretical side, moving from the univariate theory to the mutivariate; studying clustering (extreme events happening very quickly aer each other); and working towards a finer understanding of dependent processes. On the more applied side, the challenges facing humanity from climate change (leading to extreme weather events, such as flooding and droughts), and modern political and societal developments (increased interconnectivity leading to sudden financial collapses, huge amounts of data, etc) have provided many openings for the use of extreme values laws.

The main goal of this project is to close the gap between two disciplines by bringing the study of the rare events of dynamical systems closer to current Extreme Value Theory. We aim at pursuing this goal mainly on two levels. On the one hand, we want to develop a theory of multivariate extremes for dynamical systems, which will be quite exploratory, since, to our knowledge, there is almost nothing done in that direction. On the other hand, we want to deepen the study of clustering of rare events specially in the case of intense clustering which is usually quantified by a parameter called the Extremal Index. In this case, there are already substantial developments from the dynamical side, where clustering of rare events has been linked with periodic behaviour of the maximal set, M, and motivated the realisation of several connections between dynamical quantities and the Extremal Index, for example. We propose to deepen the understanding of clustering and motivated by that to obtain further connections between dynamical objects and the dependence structures studied in multivariate extremes, which will be undoubtedly ground-breaking.

The work plan presented is mainly theoretically driven but we expect it to find an echo in applied contexts, especially in climate dynamics where dynamical models are ubiquitous and the great interest in extreme events makes this area the natural testing ground.