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 k of 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.