Journal
JOURNAL OF DIFFERENTIAL EQUATIONS
Volume 253, Issue 5, Pages 1422-1438Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2012.05.016
Keywords
Random dynamical system; Nonautonomous process; Mean-square random attractor; Ultimate boundedness; Stochastic differential equations with nonlocal sample dependence; Stochastic partial differential equation
Categories
Funding
- DFG [KL 1203/7-1, LO 273/5-1]
- Spanish Ministerio de Ciencia e Innovacion [MTM2011-22411]
- Consejeria de Innovacion, Ciencia y Empresa (Junta de Andalucia) [Ayuda 2009/FQM314]
- Proyecto de Excelencia [P07-FQM-02468]
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The classical theory of random dynamical systems is a pathwise theory based on a skew-product system consisting of a measure theoretic autonomous system that represents the driving noise and a topological cocycle mapping for the state evolution. This theory does not, however, apply to nonlocal dynamics such as when the dynamics of a sample path depends on other sample paths through an expectation or when the evolution of random sets depends on nonlocal properties such as the diameter of the sets. The authors showed recently in terms of stochastic morphological evolution equations that such nonlocal random dynamics can be characterized by a deterministic two-parameter process from the theory of nonautonomous dynamical systems acting on a state space of random variables or random sets with the mean-square topology. This observation is exploited here to provide a definition of mean-square random dynamical systems and their attractors. The main difficulty in applying the theory is the lack of useful characterizations of compact sets of mean-square random variables. It is illustrated through simple but instructive examples how this can be avoided in strictly contractive cases or circumvented by using weak compactness. The existence of a pullback attractor then follows from the much more easily determined mean-square ultimate boundedness of solutions. (C) 2012 Elsevier Inc. All rights reserved.
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