  Planetesimal belts are invoked to explain the prolonged existence of debris
disks. Important parameters to model their collisional evolution and to compute
the dust production rate are the intrinsic probability of collision $P_i$ and
the mean impact velocity $U_c$. If a planet orbits close to the belt, the
values of both these parameters are affected by its secular perturbations
yielding a strong correlation between eccentricity $e$ and pericentre longitude
$\varpi$. We adopt a new algorithm to compute both $P_i$ and $U_c$ in presence
of various levels of secular correlation due to different ratios between proper
and forced eccentricity. We tested this algorithm in a standard case with a
Jupiter--sized planet orbiting inside a putative planetesimal belt finding that
it is less collisionally active compared to a self--stirred belt because of the
$e - \varpi$ coupling. The eccentricity of the planet is an important parameter
in determining the amount of dust production since the erosion rate is 10 times
faster when the planet eccentricity increases from 0.1 to 0.6. Also the initial
conditions of the belt (either warm or cold) and its average inclination
strongly affects $P_i$ and $U_c$ and then its long term collisional evolution
in presence of the planet. We finally apply our method to the planetesimal
belts supposedly refilling the dust disks around HD 38529 and $\epsilon$
Eridani. In the most collisionally active configurations, only a small fraction
of bodies smaller than 100 km are expected to be fragmented over a time--span
of 4 Gyr.
