  In this work we study the recurrence problem for quantum Markov chains, which
are quantum versions of classical Markov chains introduced by S. Gudder and
described in terms of completely positive maps. A notion of monitored
recurrence for quantum Markov chains is examined in association with Schur
functions, which codify information on the first return to some given state or
subspace. Such objects possess important factorization and decomposition
properties which allow us to obtain probabilistic results based solely on those
parts of the graph where the dynamics takes place, the so-called splitting
rules. These rules also yield an alternative to the folding trick to transform
a doubly infinite system into a semi-infinite one which doubles the number of
internal degrees of freedom. The generalization of Schur functions --so-called
FR-functions-- to the general context of closed operators in Banach spaces is
the key for the present applications to open quantum systems. An important
class of examples included in this setting are the open quantum random walks,
as described by S. Attal et al., but we will state results in terms of general
completely positive trace preserving maps. We also take the opportunity to
discuss basic results on recurrence of finite dimensional iterated quantum
channels and quantum versions of Kac's Lemma, in close association with recent
results on the subject.
