Aging and renewal events in sporadically modulated systems

We describe a form of modulation, namely a dishomogeneous Poisson process whose event rate changes sporadically and randomly in time with a chosen prescription, so as to share many statistical properties with a corresponding non-Poisson renewal process. Using our prescription the correlation function and the waiting time distribution between events are the same. If we study a continuous-time random walk, where the walker has only two possible velocities, randomly established at the times of the events, we show that the two processes also share the same second moment. However, the modulated diffusion process undergoes a dynamical transition between superstatistics and a Lévy walk process, sharing the scaling properties of the renewal process only asymptotically. The aging experiment - based on the evaluation of the waiting time for the next event, given a certain time distance between another previous event and the beginning of the observation - seems to be the key experiment to discriminate between the two processes.