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.
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