  A new class of random quantum--dynamical systems in continuous space is
introduced and studied in some detail. Each member of the class is
characterized by a Hamiltonian which is the sum of two parts. While one part is
deterministic, time--independent, and quadratic, the Weyl--Wigner symbol of the
other part is a homogeneous Gaussian random field which is delta correlated in
time and arbitrary, but smooth in position and momentum. Exact expressions for
the time evolution of both(mixed) states and observables averaged over
randomness are obtained.The differences between the quantum and the classical
behavior are clearly exhibited.As a special case it is shown that, if the
deterministic part corresponds to a particle subjected to a constant magnetic
field,the spatial variance of the averaged state grows diffusively for long
times independent of the initial state.
