  A new model of quantum random walks is introduced, on lattices as well as on
finite graphs. These quantum random walks take into account the behavior of
open quantum systems. They are the exact quantum analogues of classical Markov
chains. We explore the "quantum trajectory" point of view on these quantum
random walks, that is, we show that measuring the position of the particle
after each time- step gives rise to a classical Markov chain, on the lattice
times the state space of the particle. This quantum trajectory is a simulation
of the master equation of the quantum random walk. The physical pertinence of
such quantum random walks and the way they can be concretely realized is
discussed. Differences and connections with the already well-known quantum
random walks, such as the Hadamard random walk, are established.
