  We consider open quantum walks on a graph, and consider the random variables
defined as the passage time and number of visits to a given point of the graph.
We study in particular the probability that the passage time is finite, the
expectation of that passage time, and the expectation of the number of visits,
and discuss the notion of recurrence for open quantum walks. We also study exit
times and exit probabilities from a finite domain, and use them to solve
Dirichlet problems and to determine harmonic measures. We consider in
particular the case of irreducible open quantum walks. The results we obtain
extend those for classical Markov chains.
