State-Dependent Fractional Point Processes

Abstract

In this paper we analyse the fractional Poisson process where the state probabilitiespkνk(t),t≥ 0, are governed by time-fractional equations of order 0 < νk≤ 1 depending on the numberkof events that have occurred up to timet. We are able to obtain explicitly the Laplace transform ofpkνk(t) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on νkdiffers from that constructed from the fractional state equations (in the case of νk= ν, for allk, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally, we consider the fractional birth process governed by equations with state-dependent fractionality.