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[page 4, §1]   
[4.1.1] It follows from the general results in Ref. [1] that the first model defined by eqs. (6) and (9) is equivalent to the fractional master equation


with intitial condition


and fractional transition rates


Here the fractional time derivative D0+α,1 of order α and type 1 in eq. (10) is defined as [15]


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 ψ1t


and the Fourier transform of λr, 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 ψ1t 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 σ,τ0 was the background and motivation for the discussion in Ref. [2]. [5.2.2] It follows from eq. (1.9) in Ref. [2] by virtue of the continuity theorem [16] for characteristic functions that for the first model the continuum limit with


leads for all fixed k,u to


[5.2.3] Here the expansion cosx=1-x2/2+x4/24- has been used. [5.2.4] Therefore the solution of the first model with waiting time density ψ1t 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 ψ2t given by eq. (8). [5.3.2] In this case




From this follows


showing that the continuum limit as in eq. (17) with finite Cα does not give rise to the propagator of fractional diffusion. [5.3.3] On the other hand the conventional continuum limit with C1=limτ0σ0σ2/2dτ exists and yields


the Gaussian propagator of ordinary diffusion with diffusion constant C1.