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 L 4 and L 5 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 L 5 . 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 Beaugé and Roig (2001), and compare the results with the numerical estimations by Brož 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

We review the orbital stability of the planar circular restricted three-body problem, in the case of massless particles initially located between both massive bodies. We present new estimates of the resonance overlap criterion and the Hill stability limit, and compare their predictions with detailed dynamical maps constructed with N-body simulations. We show that the boundary between (Hill) stable and unstable orbits is not smooth but characterized by a rich structure generated by the superposition of different mean-motion resonances which does not allow for a simple global expression for stability.We propose that, for a given perturbing mass m 1 and initial eccentricity e, there are actually two critical values of the semimajor axis. All values a < a Hill are Hill-stable, while all values a > a unstable are unstable in the Hill sense. The first limit is given by the Hill-stability criterion and is a function of the eccentricity. The second limit is virtually insensitive to the initial eccentricity, and closely resembles a new resonance overlap condition (for circular orbits) developed in terms of the intersection between first and second-order mean-motion resonances.