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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

 ∂α∂⁡tα⁢p⁢r,t=Cα⁢Δ⁢p⁢r,t,0<α≤1, (1)

it has been subsequently argued in the literature that all continuous time random walks with long tailed waiting time densities ψt, i.e. with

 ψ⁢t∼t-1-α,t→∞, (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 α 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 pr,t, the probability density to find a random walker at the (discrete) lattice position rRd at time t if it started from the origin r=0 at time t=0 [9, 10]. [2.0.2] In eq. (2) the waiting time distribution ψt gives the probability density for a time interval t between two consecutive steps of the random walker, and the long time tail exponent α is the same as the order of the fractional time derivative in (1) (see [1, 2] for details). [2.0.3] As usual Δ denotes the Laplacian and the constant Cα0 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 pr,t satisfies a fractional diffusion equation” and proposition B pr,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.  and [13, p. 116ff] by showing that fractional diffusion equations of order α and type β1 (0β1) 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 α). [2.2.2] Naturally, the idea underlying the example can be widely generalized.