Markovian master equations underlie many areas of modern physics and, despite their apparent simplicity, they encode a rich and complex dynamics which is still under active research. We identify a class of continuous variable Markovian master equations for which positivity and complete positivity become equivalent. We apply this result to characterize the positivity of the partially transposed evolution of bipartite Gaussian systems, which encodes the dynamics of entanglement. Finally, the entangling properties of models of classical gravity interacting with quantum matter are investigated in the context of the experimental proposals to detect gravitationally induced entanglement. We prove that entanglement generation can indeed take place within these models. We prove that entanglement generation can indeed take place within these models. In particular, by focusing on the Diósi-Penrose model for two gravitationally interacting masses, we show that entanglement-based experiments have the potential to either falsify the model entirely or constrain the free parameter of the model $R_0$ up to values six orders of magnitude above the current state of the art.

8 pages, 4 figures, close to published version