  For the symmetric case of space-fractional diffusion processes (whose basic
analytic theory has been developed in 1952 by Feller via inversion of Riesz
potential operators) we present three random walk models discrete in space and
time. We show that for properly scaled transition to vanishing space and time
steps these models converge in distribution to the corresponding
time-parameterized stable probability distribution. Finally, we analyze in
detail a model, discrete in time but continuous in space, recently proposed by
Chechkin and Gonchar. Concerning the inversion of the Riesz potential operator
$I_0^\alpha$ let us point out that its common hyper-singular integral
representation fails for $\alpha = 1. In our Section 2 we have shown that the
corresponding hyper-singular representation for the inverse operator
$D_0^\alpha$ can be obtained also in the critical (often excluded) case $\alpha
= 1$, by analytic continuation
