Fractional Poisson processes and related planar random motions
Luisa Beghin, Universita di Roma, La Sapienza
Enzo Orsingher, Universita di Roma, La Sapienza
Abstract
We present three different fractional versions of the Poisson process and
some related results concerning the distribution of order statistics and the
compound Poisson process.
The main version is constructed by considering the difference-differential
equation governing the distribution of the standard Poisson process, $
N(t),t>0$, and by replacing the time-derivative with the fractional
Dzerbayshan-Caputo derivative of order ν ∈ ( 0,1] .
For this process, denoted by $mathcal{N}ν(t),t>0,$ we obtain an
interesting probabilistic representation in terms of a composition of the
standard Poisson process with a random time, of the form $mathcal{N}ν(t)=N(mathcal{T}_{2ν }(t)),$ $t>0$. The time argument $mathcal{T}_{2ν
}(t),t>0$, is itself a random process whose distribution is related to the
fractional diffusion equation.
We also construct a planar random motion described by a particle moving at
finite velocity and changing direction at times spaced by the fractional
Poisson process $mathcal{N}ν .$ For this model we obtain the
distributions of the random vector representing the position at time $t$,
under the condition of a fixed number of events and in the unconditional
case.
For some specific values of ν ∈ (0,1] we show that the
random position has a Brownian behavior (for ν =1/2) or a
cylindrical-wave structure (for ν =1)
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