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## IV. LAWS OF ANOMALOUS DIFFUSION 

## A. Long-time behavior of the inverse Laplace transform 

From a formal point of view, the moments 〈 Y$_{t}$ 〉 and 〈 Y 2 t 〉 completely determine the vari- ance σ $^{2}$( t ). But the calculation of the inverse Laplace transforms in Eqs. (2.13) and (2.14) is a difficult technical problem because of the contour integration in the complex plane s . Fortunately, in the long-time limit this problem can be avoided. Such a possibility provides the celebrated Tauberian theorem for the Laplace transform [18] which is widely used in the theory of CTRW and its applications. According to this theorem, if h ( t ) is ultimately monotone and 

(0 < ρ < ∞ ) as s → 0 then 

<img src='content_image/367790.jpg'>

<img src='content_image/367791.jpg'>

as t → ∞ , where L ( t ) is a slowly varying function at infinity. The term ‘slowly varying’ means that L ( λt ) ∼ L ( t ), i.e., lim$_{t}$ →∞ L ( λt ) /L ( t ) = 1, for all λ > 0. We note that, in contrast to Eqs. (2.13) and (2.14), the parameter s in (4.1) is assumed to be a positive real number. 

The Tauberian theorem in the above form permits to find only the leading terms of the asymptotic expansion of the moments 〈 Y$_{t}$ 〉 and 〈 Y 2 t 〉 as t → ∞ . If lim$_{t}$ →∞ 〈 Y$_{t}$ 〉 $^{2}$/ 〈 Y 2 t 〉 ̸ = 1 then, according to the definition (2.15), these terms completely determine also the leading term of the long-time expansion of σ $^{2}$( t ). However, if lim$_{t}$ →∞ 〈 Y$_{t}$ 〉 $^{2}$/ 〈 Y 2 t 〉 = 1 then for finding the leading term of σ $^{2}$( t ) at least the first two terms of each of the asymptotic expansions of 〈 Y$_{t}$ 〉 and 〈 Y 2 t 〉 should be evaluated. Remarkably, these terms can also be determined from the Tauberian theorem if the Laplace transforms 〈 Y$_{t}$ 〉 s and 〈 Y 2 t 〉 s at s → 0 have the asymptotic form 

<img src='content_image/367792.jpg'>

with η > ρ . In this case, replacing h$_{s}$ by h$_{s}$ − q/s η and using the exact result L − $^{1}${ 1 /s $^{η}$} = t η − $^{1}$/ Γ( η ) [19], from (4.2) we obtain 

<img src='content_image/367793.jpg'>

