content_image/375029.jpg content_image/375025.jpg content_image/375027.jpg content_image/375031.jpg content_image/375033.jpg content_image/375035.jpg content_image/375037.jpg content_image/375039.jpg
totic formula (3.2), we obtain 

as s → 0. An integration by parts together with the integral representation of the gamma function [21], Γ( x ) = ∫ ∞ 0 dy e − $^{y}$yx − $^{1}$, reduces (4.6) to the form a Γ(1 α ) 

Now, using this result and the Laplace transforms 

of the first two moments of Y$_{t}$ , we find in the limit s → 0: 

and 

Since these asymptotic formulas are particular cases of the asymptotic formula (4.1) in which the slowly varying function L (1 /s ) is a constant, from (4.2) we obtain in the long-time limit 

and 

Thus, in this case lim$_{t}$ →∞ 〈 Y$_{t}$ 〉 $^{2}$/ 〈 Y 2 t 〉 ̸ = 1 and the above asymptotic expressions yield 

<img src='content_image/375029.jpg'>

<img src='content_image/375025.jpg'>

<img src='content_image/375027.jpg'>

<img src='content_image/375031.jpg'>

<img src='content_image/375033.jpg'>

<img src='content_image/375035.jpg'>

<img src='content_image/375037.jpg'>

<img src='content_image/375039.jpg'>

According to this result, which agrees with that obtained in the context of the asymptotic solution of the CTRW [20], subdiffusion occurs if α ∈ (0 , 1 / 2) and superdiffusion if α ∈ (1 / 2 , 1). If α = 1 / 2 then σ $^{2}$( t ) ∝ t and, in accordance with the commonly used terminology, the biased diffusion is normal. However, for normal diffusion processes both the mean and variance are proportional to time. Therefore, since 〈 Y$_{t}$ 〉 ∝ t 1 / 2 at α = 1 / 2, this type of diffusion should be more appropriately termed as quasi-normal. It is also worthy to note that, according to Eqs. (3.6) and (4.13), the larger is the particle mass, the stronger is diffusion. 

