Abstract
This paper introduces a generalization of the so-called space-fractional Poisson process by extending the difference operator acting on state space present in the associated difference-differential equations to a much more general form. It turns out that this generalization can be put in relation to a specific subordination of a homogeneous Poisson process by means of a subordinator for which it is possible to express the characterizing L\'evy measure explicitly. Moreover, the law of this subordinator solves a one-sided first-order differential equation in which a particular convolution-type integral operator appears, called Prabhakar derivative. In the last section of the paper, a similar model is introduced in which the Prabhakar derivative also acts in time. In this case, too, the probability generating function of the corresponding process and the probability distribution are determined.
Highlights
Introduction and backgroundIn the last decade, it became apparent that several phenomena that can be modeled in terms of point processes are non Poissonian in nature
We introduce a generalized process by suitably time-changing a superposition of weighted space-fractional Poisson processes
It became apparent that several phenomena that can be modeled in terms of point processes are non Poissonian in nature
Summary
It became apparent that several phenomena that can be modeled in terms of point processes are non Poissonian in nature (see [2, 21] as examples). The space-fractional Poisson process is in practice a homogeneous Poisson process subordinated to an independent stable subordinator The study of weak convergence of superposition of point processes has regained interest [see e.g. 10, and the references therein] showing that different behaviours are possible Within this framework, we can consider the model we are going to describe as a weighted finite superposition of independent space-fractional Poisson processes (each of them generalizing the homogeneous Poisson process and admitting the possibility of jumps of any integer order). This generalization is based on the study of similar difference-differential equations but involving the so-called regularized Prabhakar derivative in time that generalizes the Caputo derivative in time
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