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Uniform attractors for non-autonomous random dynamical systems. (English) Zbl 1377.37038

Summary: This paper is devoted to establishing a (random) uniform attractor theory for non-autonomous random dynamical systems (NRDS). The uniform attractor is defined as the minimal compact uniformly pullback attracting random set. Nevertheless, the uniform pullback attraction in fact implies a uniform forward attraction in probability, and implies also an almost uniform pullback attraction for discrete time-sequences. Though no invariance is required by definition, the uniform attractor can have a negative semi-invariance under certain conditions.
Several existence criteria for uniform attractors are given, and the relationship between uniform and cocycle attractors is carefully studied. To overcome the measurability difficulty, the symbol space is required to be Polish which is shown fulfilled by the hulls of \(L_{l o c}^p(\mathbb{R}; L^r)\) functions, \(p, r > 1\). Moreover, uniform attractors for continuous NRDS are shown determined by uniformly attracting deterministic compact sets. Finally, the uniform attractor for a stochastic reaction-diffusion equation with translation-bounded external forcing are studied as applications.

MSC:

37C60 Nonautonomous smooth dynamical systems
37H10 Generation, random and stochastic difference and differential equations
35K57 Reaction-diffusion equations
37L55 Infinite-dimensional random dynamical systems; stochastic equations
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