Me

Limiting laws ruling dynamical systems

Reference: PTDC/MAT-PUR/28177/2017
Budget: 213.783,22 Euro
Starting date: 2018-10-04 End date: 2021-10-03

Project funded by FCT (Portugal).

Principal Investigator

Co-Principal Investigator

Team Members

Summary

In general terms, the main goal of this project is to study statistical properties of dynamical systems, which are particularly captured by the description provided by limiting laws.

The complexity of the orbital structure of chaotic systems brought special attention to the existence of such limiting laws, since they borrow at least some probabilistic predictability to the erratic behaviour of such systems.

The first step in this direction is the construction of invariant physical measures, which provide an asymptotic spatial distribution of the orbits in the phase space. Ergodicity then gives strong laws of large numbers.

The mixing properties of the system restore asymptotic independence and, in this way, allow to mimic iid processes and prove limiting laws for the mean, such as: central limit theorems, large deviation principles, invariance principles, etc.

While the limiting laws mentioned so far pertain to the mean or average behaviour of the system, in the recent years, the study of the extremal behaviour, ie, the laws that rule the appearance of abnormal observations along the orbits of the system has suffered an unprecedented development, in which the team members took a prominent role. In this field of Extreme Value Theory applied to dynamical systems, one is interested in the limiting law for the partial maxima of stochastic processes arising from the system. It turns out that this study is deeply connected with the recurrence properties to certain regions of the phase space, which is one of the most emblematic achievements of the team.

Part of the reason for the development of this field stems from the interest that Physicists working with meteorological data saw in this area. One of the great advantages of understanding the extremal behaviour of dynamical systems is that the latter often provide toy models for very complex physical phenomena. This is the case of the Lorenz equations, which show how a very simplified model for the atmospheric convection can yet capture so accurately the complex and chaotic character of weather. Hence, the study of this subject has direct applications to risk assessment associated with occurrence of meteorological abnormal phenomena.

The main purpose of this project is to carry further the study of the limiting laws for the average behaviour and the extremal behaviour of the systems, as well as to understand the connection between both.

In particular, in the extremal case, we intend to provide more information about the impact of the occurrence of extreme events and the structure of record observations, for which we plan to perform a more sophisticated analysis using tools such as the theory of point processes. Moreover, in the case of heavy tailed distributions, which in this context are associated to observables with polynomial type of singularities, the average is tied to the extremal behaviour. We intend to exploit this link and understand its consequence in the dynamical setting.

Achievements

During the project, both David Mesquita and Jorge Soares completed their PhD thesis. We make a brief review of the most relevant results and make a short description of the main achievements of the project. We use as a citation key the initials of the authors concatenated with the last two digits of the year of the publication. The details of each reference can be found below in the publications list.

  • Within the first goal of the project regarding the weak convergence of dynamically generated higher dimensional empirical Rare Events Point Processes (REPP), in [FFM19], the authors consider empirical multi-dimensional rare events point processes that keep track both of the time occurrence of extremal observations and of their severity, for stochastic processes arising from a dynamical system, by evaluating a given potential along its orbits. This is done both in the absence and presence of clustering. A new formula for the piling of points on the vertical direction of bi-dimensional limiting point processes, in the presence of clustering, is given, which is then generalised for higher dimensions. The limiting multi-dimensional processes are computed for systems with sufficiently fast decay of correlations. The complete convergence results are used to study the effect of clustering on the convergence of extremal processes, record time, and record values point processes.
  • In [BLR19], the authors study the behaviour of the shortest distance between orbits and show that under some rapidly mixing conditions, the decay of the shortest distance depends on the correlation dimension. For irrational rotations, the authors prove a different behaviour depending on the irrational exponent of the angle of the rotation. For random processes, this problem corresponds to the longest common substring problem. The result of Arratia and Waterman on sequence matching to alpha-mixing processes with exponential decay is extended.

Papers of the core of the project

  1. M. Abadi, A.C.M. Freitas, J.M. Freitas, Clustering indices and decay of correlations in non-Markovian models, Nonlinearity, 32, no. 12, 4853-4870, 2019.

  2. J. F. Alves, V. Ramos, J. Siqueira, Equilibrium stability for non-uniformly hyperbolic systems. Ergodic Theory Dynam. Systems, 39, no. 10, 2619-2642, 2019.

  3. J.F. Alves, M. A. Khan, Statistical instability for contracting Lorenz flows, Nonlinearity, 32, no. 11, 4413-4444, 2019

  4. V. Barros, L. Liao, J. Rousseau, On the shortest distance between orbits and the longest common substring problem, Advances in Mathematics, 344, 311-339, 2019.

  5. A.C.M. Freitas, J.M. Freitas, and M. Magalhães. Complete convergence and records for dynamically generated stochastic processes, Trans. Amer. Math. Soc., 373,no. 1, 435–478, 2020.

MeMeMe Me