  We generalize the concept of basin of attraction of a stable state in order
to facilitate the analysis of dynamical systems with noise and to assess
stability properties of metastable states and long transients. To this end we
examine the notions of mean sojourn times and absorption probabilities for
Markov chains and study their relation to the basins of attraction. Our
approach is applicable to a large variety of problems since in most cases the
transfer operator associated to a dynamical system can be approximated by a
Markov chain.
