State-Dependent Fractional Point Processes

Abstract

In this paper we analyse the fractional Poisson process where the state probabilities p _k ^{ν _k} ( t), t≥ 0, are governed by time-fractional equations of order 0 < ν _k ≤ 1 depending on the number kof events that have occurred up to time t. We are able to obtain explicitly the Laplace transform of p _k ^{ν _k} ( t) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on ν _k differs from that constructed from the fractional state equations (in the case of ν _k = ν, for all k, 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.