  We present a detailed study of a simple quantum stochastic process, the
quantum phase space Brownian motion, which we obtain as the Markovian limit of
a simple model of open quantum system. We show that this physical description
of the process allows us to specify and to construct the dilation of the
quantum dynamical maps, including conditional quantum expectations. The quantum
phase space Brownian motion possesses many properties similar to that of the
classical Brownian motion, notably its increments are independent and
identically distributed. Possible applications to dissipative phenomena in the
quantum Hall effect are suggested.
