  Asteroid families are groups of minor bodies produced by high-velocity
collisions. After the initial dispersions of the parent bodies fragments, their
orbits evolve because of several gravitational and non-gravitational
effects,such as diffusion in mean-motion resonances, Yarkovsky and YORP
effects, close encounters of collisions, etc. The subsequent dynamical
evolution of asteroid family members may cause some of the original fragments
to travel beyond the conventional limits of the asteroid family. Eventually,
the whole family will dynamically disperse and no longer be recognizable.
  A natural question that may arise concerns the timescales for dispersion of
large families. In particular, what is the oldest still recognizable family in
the main belt? Are there any families that may date from the late stages of the
Late Heavy Bombardment and that could provide clues on our understanding of the
primitive Solar System? In this work, we investigate the dynamical stability of
seven of the allegedly oldest families in the asteroid main belt. Our results
show that none of the seven studied families has a nominally mean estimated age
older than 2.7 Gyr, assuming standard values for the parameters describing the
strength of the Yarkovsky force. Most "paleo-families" that formed between 2.7
and 3.8 Gyr would be characterized by a very shallow size-frequency
distribution, and could be recognizable only if located in a dynamically less
active region (such as that of the Koronis family). V-type asteroids in the
central main belt could be compatible with a formation from a paleo-Eunomia
family.
