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1 Introduction

[page 1, §1]   
[1.1.1] Given the connection (established in [1, 2]) between continuous time random walks (CTRW) and diffusion equations with fractional time derivative

\frac{\partial^{\alpha}}{\partial t^{\alpha}}p(\mathbf{r},t)=C_{\alpha}\;\Delta p(\mathbf{r},t),\qquad 0<\alpha\leq 1, (1)

it has been subsequently argued in the literature that all continuous time random walks with long tailed waiting time densities \psi(t), i.e. with

\psi(t)\sim t^{{-1-\alpha}},\qquad t\to\infty, (2)

are in some sense asymptotically equivalent to a fractional diffusion equation [3, 4, 5, 6, 7, 8]. [1.1.2] Let me first explain the symbols in these two equations. [1.1.3] Of course the fractional time derivative of order \alpha in (1) is only a symbolic notation (a [page 2, §0]   definition is given in eq. (13) below). [2.0.1] Random walks on a lattice in continuous time are described by p(\mathbf{r},t), the probability density to find a random walker at the (discrete) lattice position \mathbf{r}\in\mathbb{R}^{d} at time t if it started from the origin \mathbf{r}=\mathbf{0} at time t=0 [9, 10]. [2.0.2] In eq. (2) the waiting time distribution \psi(t) gives the probability density for a time interval t between two consecutive steps of the random walker, and the long time tail exponent \alpha is the same as the order of the fractional time derivative in (1) (see [1, 2] for details). [2.0.3] As usual \Delta denotes the Laplacian and the constant C_{\alpha}\geq 0 denotes the fractional diffusion coefficient.

[2.1.1] Despite early doubts, formulated e.g. in [11, p. 78], many authors [3, 4, 5, 6, 7, 8] consider it now an established fact that proposition A p(\mathbf{r},t) satisfies a fractional diffusion equation” and proposition B p(\mathbf{r},t) is the solution of a CTRW with long time tail” are in some sense asymptotically equivalent. [2.1.2] Equivalence between propositions A and B requires that A implies B and further that B implies A. [2.1.3] One implication, namely that A implies B, was shown to be false in Refs. [12] and [13, p. 116ff] by showing that fractional diffusion equations of order \alpha and type \beta\neq 1 (0\leq\beta\leq 1) do not have a probabilistic interpretation.

[2.2.1] In this paper an example of a CTRW is given whose waiting time density fulfills eq. (2) but whose asymptotic continuum limit is not the fractional diffusion equation (1) (with the same \alpha). [2.2.2] Naturally, the idea underlying the example can be widely generalized.