  We study the dynamics of the restricted planar three-body problem near mean
motion resonances, i.e. a resonance involving the Keplerian periods of the two
lighter bodies revolving around the most massive one. This problem is often
used to model Sun--Jupiter--asteroid systems. For the primaries (Sun and
Jupiter), we pick a realistic mass ratio $\mu=10^{-3}$ and a small eccentricity
$e_0>0$. The main result is a construction of a variety of non local diffusing
orbits which show a drastic change of the osculating (instant) eccentricity of
the asteroid, while the osculating semi major axis is kept almost constant. The
proof relies on the careful analysis of the circular problem, which has a
hyperbolic structure, but for which diffusion is prevented by KAM tori. We
verify certain non-degeneracy conditions numerically.
  Based on the work of Treschev, it is natural to conjecture that diffusion
time for this problem is $\sim \frac{-\ln (\mu e_0)}{\mu^{3/2} e_0}$. We expect
our instability mechanism to apply to realistic values of $e_0$ and we give
heuristic arguments in its favor. If so, the applicability of Nekhoroshev
theory to the three-body problem as well as the long time stability become
questionable.
  It is well known that, in the Asteroid Belt, located between the orbits of
Mars and Jupiter, the distribution of asteroids has the so-called Kirkwood gaps
exactly at mean motion resonances of low order. Our mechanism gives a possible
explanation of their existence. To relate the existence of Kirkwood gaps with
Arnold diffusion, we state a conjecture on its existence for a typical
$\eps$-perturbation of the product of a pendulum and a rotator. Namely, we
predict that a positive conditional measure of initial conditions concentrated
in the main resonance exhibits Arnold diffusion on time scales $\frac{- \ln
\eps}{\eps^{2}}$.
