Positivity and complete positivity in Markovian Gaussian open quantum systems

The proper characterization of positivity of non-unitary dynamical evolutions is a notoriously difficult problem and its relation and boundary with the stronger notion of complete positivity is far from understood. However, for a particular subset of generators of markovian evolution for a gaussian N-mode system we are able to show that complete positivity and the weaker notion of positivity are actually equivalent.