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reduce it to a continuous-time random walk (CTRW) on a semi-infinite chain, and calculate the first two moments of the particle position. The connection between the waiting time probability density and the particle mass and the probability density of the random force is also presented in this section. The conditions providing the anomalous behavior of biased diffusion are formulated in Sec. III. In Sec. IV, using the Tauberian theorem and its modified version, we derive the laws of anomalous diffusion and analyze the influence of the particle mass and the random force distribution. Finally, in Sec. V we summarize our results. 

## II. MODEL AND BASIC EQUATIONS 

We consider the one-dimensional propagation of a particle in a medium composed by the layers of a fixed width l whose transport properties are assumed to be random. The motion of a particle in this medium occurs under the action of a constant external force f ( > 0), and the influence of the layers on the particle dynamics is modeled by a random force g ( x ). We assume that g ( x ) (i) is a bounded function, i.e., g ( x ) ∈ [ − g$_{0}$, g$_{0}$ ], (ii) possesses a symmetry property, i.e., g ( x ) and − g ( x ) are statistically equivalent, and (iii) has statistically independent values on different intervals of the length l . In accordance with these conditions, we approximate g ( x ) by a piecewise constant random force (see Fig. 1) whose values are distributed with the same probability density u ( g ). In this stage, we consider u ( g ) as an arbitrary symmetric probability density, u ( g ) = u ( − g ), satisfying only the normalization condition ∫ g$_{0}$ − $_{g$_{0}$}$dg u ( g ) = 1. Since the total force acting on a particle equals f + g ( X$_{t}$ ), where X$_{t}$ ( X$_{0}$ = ˙ X$_{0}$ = 0) is 

the particle position, its dynamics can be described by the motion equation 

<img src='content_image/340283.jpg'>

with µ and ν being the particle mass and the damping coefficient, respectively. According to this equation, if f > g$_{0}$ then f + g ( X$_{t}$ ) > 0 and particles can be transported to an arbitrary large distance in the positive direction of the axis x . Thus, in this case the condition X$_{t}$ | t →∞ → ∞ holds for all sample paths of g ( x ). On the contrary, if f < g$_{0}$ then for each sample path of g ( x ) there always exists a certain point L = ln$_{st}$ ( n$_{st}$ = n , n = 0 , 1 , . . . ), which is characterized by the conditions f > − g ( x ) | x<L and f < − g ( x ) | x = L $_{+0}$, where particles are stopped. The probability W$_{n}$ that n$_{st}$ = n is expressed through the 

