[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∂tαpr,t=CαΔpr,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

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 r∈Rd 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 Cα≥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 “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. [12] and [13, p. 116ff]
by showing that fractional diffusion equations of order
α and type β≠1 (0≤β≤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 α).
[2.2.2] Naturally, the idea underlying the example can
be widely generalized.