Non-Markovian master equations from piecewise dynamics

We construct a large class of completely positive and trace preserving non-Markovian dynamical maps for an open quantum system. These maps arise from a piecewise dynamics characterized by a continuous time evolution interrupted by jumps, randomly distributed in time and described by a quantum channel. The state of the open system is shown to obey a closed evolution equation, given by a master equation with a memory kernel and a inhomogeneous term. The non-Markovianity of the obtained dynamics is explicitly assessed studying the behavior of the distinguishability of two different initial system's states with elapsing time.