Stochastic representation of a class of non-Markovian completely positive evolutions

By modeling the interaction of an open quantum system with its environment through a natural generalization of the classical concept of continuous time random walk, we derive and characterize a class of non-Markovian master equations whose solution is a completely positive map. The structure of these master equations is associated with a random renewal process where each event consist in the application of a superoperator over a density matrix. Strong nonexponential decay arise by choosing different statistics of the renewal process. As examples we analyze the stochastic and averaged dynamics of simple systems that admit an analytical solution. The problem of positivity in quantum master equations induced by memory effects [ S. M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001) ] is clarified in this context.