Fractional Poisson and fractional birth processes

In this talk we consider different generalizations of the Poisson process. In particular we study the time-fractional Poisson process $N_\nu (t)$ with two different approaches, obtaining explicitly the distribution $\mathbf P[N_\nu (t)=k]$, $0<\nu <1$, $t>0$, $k\geq 0$.
We obtain the subordination relationship $N_\nu (t)=N(T_{2\nu}(t))$ where $N$ is the homogeneous Poisson process and $T_{2\nu} (t)$ is a time-process whose distribution is related to the fractional diffusion equation. We present also the fractional non-linear (and linear) birth process of which the distribution, the main moments and the structure are examined.