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Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations. (English) Zbl 1188.37076

The authors introduce the notion of quasi-continuity for a dynamical system φ on a Banach space B to mean that for any sequence (t n ,x n ) converging to (t,x) in [0,)×B, for which φ (t n ) x n n is bounded, φ(t n )x n converges weakly to φ(t)x. This property is weaker than norm-to-weak continuity. The authors invoke this notion for random dynamical systems on a separable B in order to derive a sufficient condition for the existence of a random attractor, using asymptotic compactness properties formulated in terms of the Kuratowski measure of non-compactness.

The result is applied to stochastic reaction-diffusion equations on D d bounded, with a finite-dimensional additive Wiener process taking values in L (D). The nonlinearity of the drift being polynomially bounded, one has to consider solutions in L p (D) for suitable p>2. The induced random dynamical system is quasi-continuous, but it fails to be norm-to-weak continuous. It is shown to have a finite-dimensional random attractor A for bounded subsets of L 2 (D), where attraction holds with respect to the L p (D) norm for every p2. Finally, a comparison estimate for the fractal dimensions of A with respect to L p (D) norms, p2, is given.

37L55Infinite-dimensional random dynamical systems; stochastic equations
35B41Attractors (PDE)
35R60PDEs with randomness, stochastic PDE
37L30Attractors and their dimensions, Lyapunov exponents
60H15Stochastic partial differential equations
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