  Using quantum parallelism on random walks as original seed, we introduce new
quantum stochastic processes, the open quantum Brownian motions. They describe
the behaviors of quantum walkers -- with internal degrees of freedom which
serve as random gyroscopes -- interacting with series of probes. These
processes may also be viewed as the scaling limit of open quantum random walks
and we develop this approach along three different lines: quantum trajectory,
quantum dynamical map, and quantum stochastic differential equation. We also
present a study of the simplest case, with a two level system as internal
gyroscope, illustrating the interplay between ballistic and diffusive behaviors
at work in these processes.
