What would be a fractional Poisson Process like I think that the definition of fractional Brownian Motion is widely known (for example as a Gaussian Process with particular variance covariance stucture parametrized by the so-called Hurst index). 
Heuristically, you can think of those processes as Gaussian processes with long (or short) memory depending on the value of their Hurst Index, and for Hurst index equal to 1/2 you get classical Brownian Motion (which has no memory).
I was wondering what would be the definition for "fractional Poisson Processes" and what stylised facts about fractional Brownian Motion one should consider in extending the definition to Poisson process. 
If any reference exists about this, this is just fine for me.
I have no other motivation than curiosity on this topic. 
 A: Here is a thesis containing (in Section 2) an overview of different definitions of fPP. 
My personal favorite is the "Standard Fractional generalization I" defined in 2.2. The reason is that there seems to be (I failed to find any relevant results) an isomorphism between this version and fBm similar to the (usual) Wiener-Poisson isomorphism.
A: A standard Poisson process is a renewal process with exponential distributed waiting times. Fractional Poisson process (FPP) is also a renewal process with Mittag-Leffler waiting times. Note that Mittag-Leffler distribution is a heavy tailed generalization of exponential distribution. Further, let N(t) be a standard Poisson process and $E_{\alpha}(t) = \inf\{s \geq 0: S_{\alpha}(s)>t\}$ be the first-exit time of a stable subordinator $S_{\alpha}(t)$ then the time-changed process $N^*(t) = N(E_{\alpha}(t))$ is also a characterization of FPP. 
A: Hello Everyone,
To find the definition of Fractional Poisson Process and its first two moments as well as the Compound Fractional Poisson Process go to http://pi.314159.ru/laskin3.pdf
To find more on the topic you may Google with key words: fractional Poisson distribution.
I hope it helps.
A: I came across two possibilities when it comes to introducing fractionality:
1) Subordination by inverse $\alpha$-stable subordinator $(E_\alpha(t))$ (see for example [1]): $N(E_\alpha(t))$ is equivalently derived from the renewal representation of the Poisson process and replaces the exponential distribution of the waiting times by the Mittag-Leffler distribution.
Moreover, the inverse subordinator method can also be applied to Brownian motion, which gives $B(E_\alpha(t))$. This stochastic process is self-similar, but does no longer have stationary increments as $(E_\alpha(t))$ has not.
2) Let's assume we have the fBM as defined in on Wiki. The suggested fPP in [2] starts with the Weyl integral representation of fBM and replaces the integration w.r.t. Brownian motion by integration w.r.t. a Poisson process. In this case both fBM and fPP are self-similar and have stationary increments.
References:
[1] Meerschaert, M. M. and P. Straka (2013). Inverse stable subordinators. Math. Model. Nat. Phenom. 8 (2), 1–16.
[2] Wang, X.-T., Z.-X. Wen, and S.-Y. Zhang (2006). Fractional Poisson process. II. Chaos Solitons Fractals 28 (1), 143–147.
