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Then, taking into account that l = ∫ τ ( n ) 0 d (Δ t ) v ( n $^{)}$(Δ t ), with the help of Eqs. (2.5) and (2.6) we find ( n ) ( n ) ( n ) 

<img src='content_image/357458.jpg'>

Since in the case under consideration κτ ( n ) ≫ 1, the exponential term in Eq. (2.6) can be neglected yielding v ( n ) + = ( f + g ( n $^{)}$) /ν . According to this approximation, the particle velocity at the end of the n th interval is determined by the random force on this interval. Therefore, using the continuity condition for the particle velocity, v ( n ) − = v ( n − 1) + , we obtain v ( n ) − = ( f + g ( n − $^{1)}$) /ν . Substituting these expressions for v ( n ) + and v ( n ) − into Eq. (2.7), we arrive to the following result: 

<img src='content_image/357459.jpg'>

It shows that in the weakly underdamped regime the waiting time τ ( n ) depends not only on the random force g ( n $^{)}$, as in the overdamped case, but also on the random force g ( n − $^{1)}$. Since these forces are statistically independent, the probability density of the waiting time can be written in the form 

<img src='content_image/357460.jpg'>

if τ ∈ [ τ$_{min}$, τ$_{max}$ ], otherwise it equals zero. It is not difficult to verify that in the overdamped case (when κ = ∞ ) Eq. (2.9) reduces to Eq. (2.3). 

Next, we express the first two moments of the random variable Y$_{t}$ through the waiting time probability density p ( τ ). Since the moments of N ( t ) are known from the CTRW theory (see, e.g., Ref. [16]), we reproduce here only the main results related to our situation. Introducing the probability P ( n, t ) that N ( t ) = n , we define the k th moment of the particle position Y$_{t}$ in the usual way: ∞ 

( k = 1 , 2 , . . . ). Then, using the Laplace transform of a function h ( t ), h$_{s}$ = L{ h ( t ) } = ∫ ∞ 0 dt e − $^{st}$h ( t ) (Re s > 0), and taking into account that P (0 , t ) = ∫ ∞ t dτ p ( τ ) and t 

( n ≥ 1), we obtain Y k 

<img src='content_image/357461.jpg'>

<img src='content_image/357464.jpg'>

<img src='content_image/357462.jpg'>

