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Main Title: Connectedness of random set attractors
Author(s): Scheutzow, Michael
Vorkastner, Isabell
Type: Article
Language Code: en
Abstract: We examine the question whether random set attractors for continuous-time random dynamical systems on a connected state space are connected. In the deterministic case, these attractors are known to be connected. In the probabilistic setup, however, connectedness has only been shown under stronger connectedness assumptions on the state space. Under a weak continuity condition on the random dynamical system we prove connectedness of the pullback attractor on a connected space. Additionally, we provide an example of a weak random set attractor of a random dynamical system with even more restrictive continuity assumptions on an even path-connected space which even attracts all bounded sets and which is not connected. On the way to proving connectedness of a pullback attractor we prove a lemma which may be of independent interest and which holds without the assumption that the state space is connected. It states that even though pullback convergence to the attractor allows for exceptional nullsets which may depend on the compact set, these nullsets can be chosen independently of the compact set (which is clear for σ-compact spaces but not at all clear for spaces which are not σ-compact).
Issue Date: 28-Dec-2018
Date Available: 7-Mar-2019
DDC Class: 510 Mathematik
Subject(s): random dynamical system
pullback attractor
weak attractor
pullback continuity
measurable selection
Journal Title: Rendiconti Lincei - Matematica E Applicazioni
Publisher: European Mathematical Society (EMS)
Publisher Place: Zürich
Volume: 29
Issue: 4
Publisher DOI: 10.4171/RLM/824
Page Start: 607
Page End: 617
EISSN: 1720-0768
ISSN: 1120-6330
Notes: Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.
This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.
Appears in Collections:FG Stochastische Analysis » Publications

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