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

Collisional parameters of planetesimal belts, precursor of debris disks, perturbed by a nearby giant planet