  We consider a model of open quantum random walk and together with a quantum
trajectory approach we are able to examine a notion of hitting time. We see
that many constructions, such as minimal solutions to hitting time problems,
are variations of well-known classical probability results, but the density
matrix degree of freedom on each site gives rise to systems which are seen to
be nonclassical. As a more specific application we study the collection of
walks induced by normal commuting contractions, for which the corresponding
probability expressions are obtained. We examine quantum versions of the
gambler's ruin, birth-and-death chain and a basic theorem on potential theory.
