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      A Generalization of the Space-Fractional Poisson Process and its Connection to some L\'evy Processes

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          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.

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          Author and article information

          Journal
          2015-02-10
          2016-01-08
          Article
          10.1214/16-ECP4383
          1502.03115
          18c2bcea-7333-4e30-9304-16c2e25dca96

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          Electronic Communications in Probability, Vol. 21, art. 20, 2016
          math.PR

          Probability
          Probability

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