Aims. In this paper, we analyze the collisional and dynamical evolution of the population of L<SUB>4</SUB> Jovian Trojans. Methods. To do this, we test different collisional parameters and include a dynamical treatment, taking into account the stability and instability zones of the L<SUB>4</SUB> Trojan swarm. This procedure allows us to estimate the size distribution of the L<SUB>4</SUB> Trojans, to study their mean collisional lifetimes, to analyze the formation of families, to obtain ejection rates of Trojan fragments and to discuss their possible contribution to the current populations of Centaurs and Jupiter-family comets. Results. Our estimates of the L<SUB>4</SUB> Trojan cumulative size distribution show waves that propagate from diameters of ∼0.1 to ∼80 km around the values derived from optical surveys. On the other hand, the mean collisional lifetimes obtained from our simulations indicate that the large Trojan asteroids have likely survived without being catastrophically fragmented over the age of the Solar System. With regards to the Trojan removal, we calculate a maximum ejection rate of Trojan fragments from L<SUB>4</SUB> of ∼50 objects larger than 1 km of diameter per Myr, which results to be significantly smaller than values previously published. Such estimates allow us to infer that the contribution of the Trojan asteroids to the current populations of Centaurs and Jupiter-family comets is negligible. In addition, our results are in agreement with the formation of few Trojan families in the L<SUB>4</SUB> swarm. On the other hand, we infer that the current orbital distribution of the Trojan asteroids does not offer a strong constraint on the dynamical origin of this population.Facultad de Ciencias Astronómicas y GeofísicasInstituto de Astrofísica de La Plata