Additive processes are obtained from Levy ones by relaxing the condition of stationary increments, hence they are spatially (but not temporally) homogeneous. By analogy with the case of time-homogeneous Markov processes, one can de ne an in nitesimal generator, which is, of course, a timedependent operator. Additive versions of stable and Gamma processes have been considered in the literature.
The space-fractional and the time-fractional Poisson processes are two well-known models of fractional evolution. They can be constructed as standard Poisson processes with the time variable replaced by a stable subordinator and its inverse, respectively. The aim of this paper is to study nonhomogeneous versions of such models, which can be defined by means of the so-called multistable subordinator (a jump process with nonstationary increments), denoted by H:=H(t), t≥0.
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