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 $\nu\in(0,1]$. For this process, denoted by $\mathcal{N}_\nu(t),t>0,$ we obtain an interesting probabilistic representation in terms of a composition of the standard Poisson process with a random time, of the form $\mathcal{N}_\nu(t)= N(\mathcal{T}_{2\nu}(t)),$ $t>0$. The time argument $\mathcal{T}_{2\nu }(t),t>0$, 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 $\mathcal{N}_\nu.$ 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 $\nu\in(0,1]$ we show that the random position has a Brownian behavior (for $\nu =1/2$) or a cylindrical-wave structure (for $\nu =1$).

In this paper the solutions u_ν=u_ν(x, t) to fractional diffusion equations of order 0<ν≤2 are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order ν=1/2ⁿ, n≥1, we show that the solutions u_1/2ⁿ correspond to the distribution of the n-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order ν=2/3ⁿ, n≥1, is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that u_ν coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions u_ν and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.

We consider some fractional extensions of the recursive differential equation governing the Poisson process, i.e. $\partial_tp_k(t)=-\lambda(p_k(t)-p_{k-1}(t))$, $k\geq0$, $t>0$ by introducing fractional time-derivatives of order $\nu,2\nu,\ldots,n\nu$. We show that the so-called "Generalized Mittag-Leffler functions" $E_{\alpha,\beta^k}(x)$, $x\in\mathbb{R}$ (introduced by Prabhakar [24] )arise as solutions of these equations. The corresponding processes are proved to be renewal, with density of the intearrival times (represented by Mittag-Leffler functions) possessing power, instead of exponential, decay, for $t\to\infty$. On the other hand, near the origin the behavior of the law of the interarrival times drastically changes for the parameter $\nu$ varying in $(0,1]$. For integer values of $\nu$, these models can be viewed as a higher-order Poisson processes, connected with the standard case by simple and explict relationships.