it has been subsequently argued in the literature that all continuous time random walks with long tailed waiting time densities , 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 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 , the probability density to find a random walker at the (discrete) lattice position at time if it started from the origin at time [9, 10]. [2.0.2] In eq. (2) the waiting time distribution gives the probability density for a time interval between two consecutive steps of the random walker, and the long time tail exponent is the same as the order of the fractional time derivative in (1) (see [1, 2] for details). [2.0.3] As usual denotes the Laplacian and the constant 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 “ satisfies a fractional diffusion equation” and proposition B “ 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 and type () do not have a probabilistic interpretation.