  Open Quantum Walks (OQWs), originally introduced by S. Attal, are quantum
generalizations of classical Markov chains. Recently, natural continuous time
models of OQW have been developed by C. Pellegrini. These models, called
Continuous Time Open Quantum Walks (CTOQWs), appear as natural continuous time
limits of discrete time OQWs. In particular they are quantum extensions of
continuous time Markov chains. This article is devoted to the study of
homogeneous CTOQW on $\mathbb{Z}^d$. We focus namely on their associated
quantum trajectories which allow us to prove a Central Limit Theorem for the
"position" of the walker as well as a Large Deviation Principle.
