  We present a series of numerical integrations of observed and fictitious
Jupiter Trojan asteroids, under the gravitational effects of the four outer
planets, for time-spans comparable with the age of the Solar System. From these
results we calculate the escape rate from each Lagrange point, and construct
dynamical maps of "permanence" time in different regions of the phase space.
Fictitious asteroids in L4 and L5 show no significant difference, showing
almost identical dynamical maps and escape rates. For real Trojans, however, we
found that approximately 23% o f the members of the leading swarm escaped after
4.5 Gyrs, while this number increased to 28.3% for L5. This implies that the
asymmetry between the two populations increases with time, indicating that it
may have been smaller at the time of formation/capture of these asteroids.
Nevertheless, the difference in chaotic diffusion cannot, in itself, account
for the current observed asymmetry (~40%), and must be primarily primordial and
characteristic of the capture mechanism of the Trojans. Finally, we calculate
new proper elements for all the numbered Trojans using the semi-analytical
approach of Beaug\'e and Roig (2001), and compare the results with the
numerical estimations by Bro\v{z} and Rosehnal (2011). For asteroids that were
already numbered in 2011, both methods yield very similar results, while
significant differences were found for those bodies that became numbered after
2011.
