GLYPH<gl19> GLYPH<gl19> GLYPH<gl24> GLYPH<gl23> GLYPH<gl22> GLYPH<gl21> GLYPH<gl79> GLYPH<gl79> GLYPH<gl79> GLYPH<gl79> GLYPH<gl79> GLYPH<gl91> GLYPH<gl74> GLYPH<gl19> GLYPH<gl16>GLYPH<gl74> GLYPH<gl74>GLYPH<gl11>GLYPH<gl91>GLYPH<gl12> FIG. 1: Sample path of a piecewise constant random force g ( x ). probability I = ∫ f − $_{g$_{0}$}$dg u ( g ) that g ( x ) < f as follows: W$_{n}$ = I $^{n}$− I n $^{+1}$. Therefore, the average distance 〈 L 〉 (the angular brackets denote an average over the sample paths of g ( x )) from the origin to the stopping point can be written in the form 〈 L 〉 = $^{l}$∑ ∞ n $_{=1}$nW$_{n}$ = $^{l}$∑ ∞ n $_{=1}$I $^{n}$. Finally, using the geometric series formula, we obtain the desired result 〈 L 〉 = l I 1 − I . (2.2) Assuming that the probability that | g ( x ) | = g$_{0}$ equals zero, i.e., the probability density u ( g ) is not concentrated at the edges of the interval [ − g$_{0}$, g$_{0}$ ], one can make sure that I → 1 and so 〈 L 〉 → ∞ as f → g$_{0}$ . Hence, at f = g$_{0}$ the condition X$_{t}$ | t →∞ → ∞ holds almost surely, i.e., with probability one. In contrast, if the probability density u ( g ) has unbounded support with g$_{0}$ = ∞ as, e.g., for a Gaussian distribution, then I < 1 and so 〈 L 〉 is finite for all finite values of the driving force f . In other words, in this case particles cannot be transported to an arbitrary large distance. It is therefore we consider here only a class of probability densities u ( g ) with bounded support. It should be noted in this context that, since infinite values of g ( x ) are physically not relevant, the assumption of bounded support is not too restrictive. Our aim is to study the long-time behavior of the particle position X$_{t}$ at f ≥ g$_{0}$ . The main statistical characteristic of X$_{t}$ is its probability density function P ( x, t ) defined as P ( x, t ) = 〈 δ ( x − X$_{t}$ ) 〉 , where δ ( x ) is the Dirac δ function. If the solution of Eq. (2.1) were known for all sample paths of g ( x ), it would be, in principle, possible to determine P ( x, t ) directly from the definition. However, this approach is difficult to implement and, what is more important, it is not necessary for finding the long-time behavior of the moments of 4