  It is known that the dynamical evolution of a system, from an initial tensor
product state of system and environment, to any two later times, t1,t2 (t2>t1),
are both completely positive (CP) but in the intermediate times between t1 and
t2 it need not be CP. This reveals the key to the Markov (if CP) and nonMarkov
(if it is not CP) avataras of the intermediate dynamics. This is brought out
here in terms of the quantum stochastic map A and the associated dynamical map
B -- without resorting to master equation approaches. We investigate these
features with four examples which have entirely different physical origins (i)
a two qubit Werner state map with time dependent noise parameter (ii)
Phenomenological model of a recent optical experiment (Nature Physics, 7, 931
(2011)) on the open system evolution of photon polarization. (iii) Hamiltonian
dynamics of a qubit coupled to a bath of $N$ qubits and (iv) two qubit unitary
dynamics of Jordan et. al. (Phys. Rev. A 70, 052110 (2004)) with initial
product states of qubits. In all these models, it is shown that the
positivity/negativity of the eigenvalues of intermediate time dynamical B map
determines the Markov/non-Markov nature of the dynamics.
