  We investigate the Markovian and non-Markovian dynamics of Gaussian quantum
channels, exploiting a recently introduced necessary and sufficient criterion
and the ensuing measure of non-Markovianity based on the violation of the
divisibility property of the dynamical map. We compare the paradigmatic
instances of Quantum Brownian motion (QBM) and Pure Damping (PD) channels, and
for the former we find that the exact dynamical evolution is always
non-Markovian in the finite-time as well as in the asymptotic regimes, for any
nonvanishing value of the non-Markovianity parameter. If one resorts to the
rotating wave approximated (RWA) form of the QBM, that neglects the anomalous
diffusion contribution to the system dynamics, we show that such approximation
fails to detect the non-Markovian nature of the dynamics. Finally, for the
exact dynamics of the QBM in the asymptotic regime, we show that the
quantifiers of non-Markovianity based on the distinguishability between quantum
states fail to detect the non-Markovian nature of the dynamics.
