Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction–diffusion equations

This paper first introduces the so-called quasi-continuous random dynamical system (RDS) on a separable Banach space. The quasi-continuity is weaker than all the usual continuities and thus is easier to check in practice. We then establish a necessary and sufficient condition for the existence of random attractors for the quasi-continuous RDS. We also give a general method to obtain the random attractors for the RDS on the Banach space Lq(D) for q⩾2. As an application, it is shown that the RDS generated by the stochastic reaction–diffusion equation possesses a finite-dimensional random attractor in Lq(D) for any q⩾2, a comparison result of fractal dimensions under the different Lq-norms is also obtained.