Diffusive behavior from a quantum master equation

We study a general class of translation invariant quantum Markov evolutions for a particle on ${\text{bbZ}}^d$. The evolution consists of free flow, interrupted by scattering events. We assume spatial locality of the scattering events and exponentially fast relaxation of the momentum distribution. It is shown that the particle position diffuses in the long time limit. This generalizes standard results about central limit theorems for classical (non-quantum) Markov processes.