  We consider generalized definitions of mixing and exactness for random
dynamical systems in terms of Markov operator cocycles. We first give six
fundamental definitions of mixing for Markov operator cocycles in view of
observations of the randomness in environments, and show that they can be
reduced into two different groups. Secondly, we give the definition of
exactness for Markov operator cocycles and show that Lin's criterion for
exactness can be naturally extended to the case of Markov operator cocycles.
Finally, in the class of asymptotically periodic Markov operator cocycles, we
show the Lasota-Mackey type equivalence between mixing, exactness and
asymptotic stability.
