with intitial condition
and fractional transition rates
thereby giving a more precise meaning to the symbolic notation in eq. (1).
[page 5, §1] [5.1.1] The result is obtained from inserting the Laplace transform of
and the Fourier transform of , the so called structure function
into eq. (5). [5.1.2] This gives
where the Fourier transform of eq. (12) was used in the last equality and the subscript refers to the first model. [5.1.3] Equation (16) equals the result obtained from Fourier-Laplace transformation of the fractional Cauchy problem defined by equations (10) and (11). [5.1.4] Hence a CTRW-model with and the fractional master equation describe the same random walk process in the sense that their fundamental solutions are the same.
[5.2.1] The continuum limit was the background and motivation for the discussion in Ref. . [5.2.2] It follows from eq. (1.9) in Ref.  by virtue of the continuity theorem  for characteristic functions that for the first model the continuum limit with
leads for all fixed to
[5.2.3] Here the expansion has been used. [5.2.4] Therefore the solution of the first model with waiting time density converges in the continuum limit to the solution of the fractional diffusion equation
with initial condition analogous to eq. (11).
[5.3.1] Consider now the second model with waiting time density given by eq. (8). [5.3.2] In this case
From this follows
showing that the continuum limit as in eq. (17) with finite does not give rise to the propagator of fractional diffusion. [5.3.3] On the other hand the conventional continuum limit with exists and yields
the Gaussian propagator of ordinary diffusion with diffusion constant .