  We present new properties for the Fractional Poisson process and the
Fractional Poisson field on the plane. A martingale characterization for
Fractional Poisson processes is given. We extend this result to Fractional
Poisson fields, obtaining some other characterizations. The fractional
differential equations are studied. We consider a more general Mixed-Fractional
Poisson process and show that this process is the stochastic solution of a
system of fractional differential-difference equations. Finally, we give some
simulations of the Fractional Poisson field on the plane.
