  We analyze in full detail the time evolution of an open Gaussian quantum
system rapidly bombarded by Gaussian ancillae. As a particular case this
analysis covers the thermalization (or not) of a harmonic oscillator coupled to
a thermal reservoir made of harmonic oscillators. We derive general results for
this scenario and apply them to the problem of thermalization. We show that
only a particular family of system-environment couplings will cause the system
to thermalize to the temperature of its environment. We discuss that if we want
to understand thermalization as ensuing from the Markovian interaction of a
system with the individual microconstituents of its (thermal) environment then
the process of thermalization is not as robust as one might expect.
