On the Exit Time from Open Sets of some Semi-Markov Processes

In this paper we characterize the distribution of the first exit time from an arbitrary open set for a class of semi-Markov processes obtained as time-changed Markov processes. We estimate the asymptotic behaviour of the survival function (for large t) and of the distribution function (for small t) and we provide some conditions for absolute continuity. We have been inspired by a problem of neurophysiology and our results are particularly useful in this field, precisely for the so-called Leaky Integrate-and-Fire (LIF) models: the use of semi-Markov processes in these models appear to be realistic under several aspects, e.g., it makes the inter-times between spikes a r.v. with infinite expectation, which is a desiderable property. Hence, after the theoretical part, we provide a LIF model based on semi-Markov processes.