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modified Tauberian theorem. It should also be noted that adding to F ( x ) a weak uniform bias breaks the symmetry of the system and, as a consequence, leads to a time dependence of 〈 x 〉 [25]. However, the biased maps considered in [25] do not exhibit anomalous diffusion at long times. 

## F. Role of thermal fluctuations 

We complete our analysis with a qualitative discussion of the role of thermal fluctuations. These fluctuations can be accounted for by adding the thermal noise term to the right-hand side of Eq. (2.1). In this case some important conclusions can be drawn from the asymptotic behavior of the correlator K ( x, y ) = 〈 [ U ( x ) − U ( y )] $^{2}$〉 as | x − y | → ∞ , where U ( x ) is the random potential that corresponds to the random force g ( x ) = − dU ( x ) /dx . Since under thermal fluctuations particles can move in both directions, the random force and potential should be determined on the entire x -axis. Using the properties of g ( x ) and the continuity condition for U ( x ) with U (0) = 0, we obtain 

<img src='content_image/337096.jpg'>

Here, x ∈ [ nl, nl + l ), n = 0 , ± 1 , . . . , and U ( nl ) = − $^{l}$∑ n s $_{=0}$g ( s ) + lg ( n ) if n ≥ 0 and U ( nl ) = $^{l}$∑ n s = − $_{1}$g ( s ) if n ≤ − 1. Taking into account that the random forces g ( s ) with different s are statistically independent, the definition of the correlator K ( x, y ) and Eq. (4.37) lead to the asymptotic expression 

( | x − y | → ∞ ), where σ 2 g = ∫ g$_{0}$ − g$_{0}$ dg g $^{2}$u ( g ). The systems with K ( x, y ) x y 

∝ | − | have long been a subject of extensive study (see, e.g., Refs. [6, 26] and references therein). A remarkable result obtained for these systems in the overdamped regime is that there always exists a threshold value f$_{tr}$ of the external force f in which the depinning transition occurs. This transition is characterized by vanishing the average particle velocity in the pinning state, when f ≤ f$_{tr}$ , while in the depinning state, when f > f$_{tr}$ , particles move with a nonzero average velocity which strongly depends on f . According to [27], if g ( x ) is a bounded function then f$_{tr}$ < g$_{0}$ ( f$_{tr}$ → g$_{0}$ as the temperature approaches zero) and the mean first-passage time in the depinning and pinning states is finite and infinite, respectively. Since the moments of the first-passage time can be associated with 

<img src='content_image/337097.jpg'>

