[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

(10) |

with intitial condition

(11) |

and fractional transition rates

(12) |

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

(13) |

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

(14) |

and the Fourier transform of , the so called structure function

(15) |

into eq. (5). [5.1.2] This gives

(16) |

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. [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

(17) |

leads for all fixed to

(18) |

[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

(19) |

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

(20) |

and

(21) | |||

From this follows

(22) |

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

(23) |

the Gaussian propagator of ordinary diffusion with diffusion constant .