We consider the issue of non-Markovianity of a quantum dynamics starting from
a comparison with the classical definition of Markovian process. We point to
the fact that two sufficient but not necessary signatures of non-Markovianity
of a classical process find their natural quantum counterpart in recently
introduced measures of quantum non-Markovianity. This behavior is analyzed in
detail for quantum dynamics which can be built taking as input a class of
classical processes.Comment: 15 pages, 6 figures; to appear in J. Phys. B, Special Issue on "Loss
of coherence and memory effects in quantum dynamics
