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Random perturbations and statistical properties of dynamical systems

Grant number: 15/02230-9
Support type:Scholarships in Brazil - Post-Doctorate
Effective date (Start): February 01, 2016
Effective date (End): September 30, 2016
Field of knowledge:Physical Sciences and Mathematics - Mathematics
Principal Investigator:Paulo Regis Caron Ruffino
Grantee:Christian da Silva Rodrigues
Home Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
Associated research grant:11/50151-0 - Dynamical phenomena in complex networks: fundamentals and applications, AP.TEM

Abstract

The study of statistical properties of Dynamical Systems is currently present in almost all fieldsof science, from fundamental Mathematics to applied modelling. One, for example, is interested in knowinghow robust the conclusions drawn from a model is. This task is particularly intricate when the dynamicsevolve under random bounded perturbations. In this case the study of statistical properties of the dynamicsis usually carried out in terms of Markov chains and/or perturbations on the space of maps. The mainchallenge is that this requires strong assumptions on the properties of the perturbations themselves, on theexistence of nearby orbits, on the space where it takes place, and on the classes of systems. Contrasting withstochastic processes under unbounded perturbations where tools from statistical analysis are applied, in thecase of bounded perturbations only some punctual progress on this direction has been made. Furthermore,there is no general approach to address some of the most fundamental questions in this field. This projectaims at studying statistical properties of some of the most important classes of dynamical systems. Inparticular, it is intended to make substantial contributions connecting structural properties with statisticalones. The issues that should be addressed include a formulation of a general criterion for stochastic stabilityin light of the possibility of representation of random maps, thoroughly established in terms of shadowingproperty, and a formulation of a general principle to study statistical stability in terms of Optimal TransportTheory and the solution of the Monge problem.