  Markov processes play an important role in physics and the theory of open
systems in particular. In this paper we study the asymptotic evolution of
trace-nonincreasing homogenous quantum Markov processes (both types, discrete
quantum Markov chains and continues quantum dynamical semigroups) equipped with
a so-called strictly positive T -state in the Schrodinger and the Heisenberg
picture. We derive a fundamental theorem specifying the structure of the
asymptotic and uncover a rich set of transformations between attractors of
quantum Markov processes in both pictures. Moreover, we generalize the
structure theorem derived for quantum Markov chains to quantum dynamical
semigroups showing how the internal structure of generators of quantum Markov
processes determines attractors in both pictures. Based on these results we
provide two characterizations of all asymptotic and stationary states, both
strongly reminding in form the well-known Gibbs states of statistical
mechanics. We prove that the dynamics within the asymptotic space is of unitary
type, i.e. quantum Markov processes preserve a certain scalar product of
operators from the asymptotic space, but there is no corresponding unitary
evolution on the original Hilbert space of pure states. Finally simple examples
illustrating the derived theory are given.
