and Therefore, in accordance with the modified Tauberian theorem, Eqs. (4.3) and (4.4), we obtain and As a consequence, the long-time behavior of the variance is described by the asymptotic formula The fact that σ $^{2}$( t ) increases faster than t is in accordance with the asymptotic formula (4.20). Indeed, while t 3 − α approaches t , the coefficient of proportionality between σ $^{2}$( t ) and t tends to infinity as α → 2 − 0. We note also that the ratio a/τ 3 at α = 2 is determined by the same Eq. (4.22). Therefore, if u ( g ) is given by Eq. (3.7) then the larger is the particle mass, the weaker is diffusion. In conclusion of this subsection we would like to draw attention to the differences between our model and one-dimensional iterated maps which generate trajectories according to the rule x$_{n}$$_{+1}$ = x$_{n}$ + F ( x$_{n}$ ). It is usually assumed [22] that F ( x ) is an antisymmetric, F ( − x ) = − F ( x ), and periodic, F ( x + N ) = F ( x ) ( N is an integer), function. Due to these conditions, there is no drift, i.e., the quantity 〈 x 〉 ≡ 〈 x$_{n}$$_{+}$$_{t}$ − x$_{n}$ 〉 can be taken to be zero, where the angular brackets denote an average over a properly chosen set of initial conditions of x$_{n}$ and t plays the role of the number of iterations. The same property, 〈 x 〉 = 0, holds also for maps perturbed by time dependent noise [23] and quenched disorder [24] with zero means. Thus, these maps are unbiased and so the variance of x$_{n}$$_{+}$$_{t}$ − x$_{n}$ , i.e., 〈 x $^{2}$〉 − 〈 x 〉 $^{2}$, reduces to 〈 x $^{2}$〉 . Using the CTRW theory, in most cases it is possible to write this variance as the inverse Laplace transform and use the ordinary Tauberian theorem for finding its long-time behavior [22, 23]. In contrast, in our model 〈 Y$_{t}$ 〉 grows with time and although the moments 〈 Y$_{t}$ 〉 and 〈 Y 2 t 〉 can also be represented as the inverse Laplace transform, the variance σ $^{2}$( t ) = 〈 Y 2 t 〉 − 〈 Y$_{t}$ 〉 2 cannot. Therefore, if lim$_{t}$ →∞ 〈 Y$_{t}$ 〉 $^{2}$/ 〈 Y 2 t 〉 = 1, the leading term of the asymptotic expansion of σ $^{2}$( t ) as t → ∞ cannot be determined by applying the ordinary Tauberian theorem. We have solved this problem (for α = 1) by using the 15 〈 Y 2 t 〉 s − 2 l 2 τ 2 1 s 3 ∼ 2 l $^{2}$a τ 3 1 s 2 ln 1 s . (4.33) 〈 Y$_{t}$ 〉 − l τ t ∼ la 2 τ 2 ln t (4.34) 〈 Y 2 t 〉 − l 2 τ 2 t 2 ∼ 2 l $^{2}$a τ 3 t ln t. (4.35) σ $^{2}$( t ) ∼ l $^{2}$a τ 3 t ln t. (4.36)