where denotes the probability for a displacement in each single step, and is the waiting time distribution giving the probability density for the time interval between two consecutive steps. [2.3.3] The transition probabilities obey , and the function is the survival probability at the initial position which is related to the waiting time distribution through
[page 3, §0] Fourier-Laplace transformation leads to the solution in Fourier-Laplace space given as 
where is the Fourier-Laplace transform of and similarly for and .
where is the characteristic time, and
is the generalized Mittag-Leffler function . [3.1.3] In the second model the waiting time density is chosen as
where , and is a suitable dimensional constant.
[3.2.1] The waiting time density differs only little from as shown graphically in Figure 1.
[3.2.2] Note that both models have a long time tail of the form given in eq. (2), and the average waiting time diverges.
[3.3.1] For both models the spatial transition probabilities are chosen as those for nearest-neighbour transitions (Polya walk) on a -dimensional hypercubic lattice given as
where is the -th unit basis vector generating the lattice, is the lattice constant and for and for .