Limiting laws ruling
dynamical systems

PTDC/MAT-PUR/28177/2017

Total budget: 213.783,22 €

Starting date: 2018-10-04. End date: 2021-10-03.

Project
funded by FCT (Portugal).

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Principal Investigator

·
Jorge Miguel Milhazes
de Freitas

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Co-Principal Investigator

·
Ana
Cristina Moreira Freitas

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Project members

·
Ana
Cristina Moreira Freitas (co-PI, FCUP)

·
Jorge
Milhazes Freitas (PI, FEP)

·
José
Ferreira Alves (FCUP)

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Mário
Magalhães (Postdoc)

·
Romain Aimino (Postdoc)

·
David Mesquita (Ph.D. student)

·
Jorge
Soares (Ph.D. student)

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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.

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Achievements:

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Papers
published within the project: