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 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 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 h$_{s}$ = L{ h ( t ) } ∼ L ( 1 s ) 1 s ρ (4.1) h ( t ) = L − $^{1}${ h$_{s}$ } ∼ 1 Γ( ρ ) L ( t ) t ρ − 1 (4.2) h$_{s}$ − q s η ∼ L ( 1 s ) 1 s ρ (4.3) h ( t ) − q Γ( η ) t η − 1 ∼ 1 Γ( ρ ) L ( t ) t ρ − $^{1}$. (4.4) 9