content_image/342138.jpg content_image/342139.jpg content_image/342140.jpg content_image/342141.jpg content_image/342142.jpg content_image/342143.jpg
necessary condition for anomalous diffusion. We note also that in this case one may expect that the rescaled probability density P ( ψ, t ) at t → ∞ approaches the stable probability density, as the generalized central limit theorem [18] suggests. 

The coefficient of proportionality a and the exponent α are in general not independent and can be found from the asymptotic behavior of the probability density u ( g ) in the vicinity of the point g = − g$_{0}$ . Indeed, using the condition ∫ g$_{0}$ − g$_{0}$ dg gu ( g ) = 0, which is a consequence of the symmetry property of u ( g ), from Eq. (2.9) at f = g$_{0}$ we obtain 

( τ → ∞ , κ → ∞ ). Then, assuming that 

<img src='content_image/342138.jpg'>

<img src='content_image/342139.jpg'>

where g → − g$_{0}$ , b > 0 and β > 0, the asymptotic formula (3.3) takes the form 

Finally, comparing (3.5) with (3.2), we find α = β and 

<img src='content_image/342140.jpg'>

<img src='content_image/342141.jpg'>

where a$_{od}$ = bg$_{0}$ ( νl/g$_{0}$ ) α is the parameter a in the overdamped limit. According to these results, the particle mass decreases the parameter a in comparison with the overdamped case but does not change the exponent α . 

An example of u ( g ) having the asymptotic behavior (3.4) is the probability density 

where Γ( x ) = ∫ ∞ 0 dy y x − $^{1}$e − y is the gamma function, which corresponds to the symmetric beta distribution. According to Eq. (3.7), this distribution is unimodal with the maximum at g = 0 if β > 1, bimodal with infinite maxima at g = ± g$_{0}$ if β < 1, and uniform if β = 1. Determining the parameter b directly from the density function (3.7), for this example we obtain α 

<img src='content_image/342142.jpg'>

<img src='content_image/342143.jpg'>

