The authors introduce the notion of quasi-continuity for a dynamical system on a Banach space to mean that for any sequence converging to in , for which is bounded, converges weakly to . This property is weaker than norm-to-weak continuity. The authors invoke this notion for random dynamical systems on a separable 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 bounded, with a finite-dimensional additive Wiener process taking values in . The nonlinearity of the drift being polynomially bounded, one has to consider solutions in for suitable . 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 for bounded subsets of , where attraction holds with respect to the norm for every . Finally, a comparison estimate for the fractal dimensions of with respect to norms, , is given.