Criteria for strong and weak random attractors

The notion of an attractor is one of the basic concepts in the theory of dynamical systems. For deterministic systems this notion has been of importance since several decades. Since about fifteen years also attractors for stochastic systems have been taken under consideration. The crucial obstacle came from the fact that the classical approach, using the Markov property of individual solutions to define the Markov semigroup and an associated generator, could not deal with joint motions of two or more points. This began to change after the introduction of stochastic flows, going back to Kunita and to Elworthy, and the introduction of the notion of random dynamical systems (see Arnold [1]). All approaches to random attractors use the theory of random dynamical systems (RDS). The notion of pullback attractors, which are referred to as strong attractors here, go back to Crauel, Flandoli, and Debussche [12, 10], and to Schmalfuß [22]. Later the notion