  In the present paper, we construct QMCs associated with Open Quantum Random
Walks such that the transition operator of the chain is defined by OQRW and the
restriction of QMC to the commutative subalgebra coincides with the
distribution $P_\rho$ of OQRW. This sheds new light on some properties of the
measure $P_\rho$. As an example, we simply mention that the measure can be
considered as a distribution of some functions of certain Markov process.
Furthermore, we study several properties of QMC and associated measure. A new
notion of $\f$-recurrence of QMC is studied, and it is established relations
between the defined recurrence and the existing ones.
