  We introduce a general construction of master equations with memory kernel
whose solutions are given by completely positive trace preserving maps. These
dynamics going beyond the Lindblad paradigm are obtained with reference to
classical renewal processes, so that they are termed quantum renewal processes.
They can be described by means of semigroup dynamics interrupted by jumps,
separated by independently distributed time intervals, following suitable
waiting time distributions. In this framework, one can further introduce
modified processes, in which the first few events follow different
distributions. A crucial role, marking an important difference with respect to
the classical case, is played by operator ordering. Indeed, for the same choice
of basic quantum transformations, different quantum dynamics arise. In
particular, for the case of modified processes, it is natural to consider the
time inverted operator ordering, in which the last few events are distributed
differently.
