  The dissipative dynamics of a quantum Brownian particle is studied for
different types of environment. We derive analytic results for the time
evolution of the mean energy of the system for Ohmic, sub-Ohmic and super-Ohmic
environments, without performing the Markovian approximation. Our results allow
to establish a direct link between the form of the environmental spectrum and
the thermalization dynamics. This in turn leads to a natural explanation of the
microscopic physical processes ruling the system time evolution both in the
short-time non-Markovian region and in the long-time Markovian one. Our
comparative study of thermalization for different environments sheds light on
the physical contexts in which non-Markovian dissipation effects are dominant.
