STATE-DEPENDENT FRACTIONAL POINT PROCESSES

In this paper we analyse the fractional Poisson process where the state probabilities p(k)(vk) (t), t >= 0, are governed by time-fractional equations of order 0 < v(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)(vk) (t) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on vk differs from that constructed from the fractional state equations (in the case of v(k) = v, 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.