  We develop a systematic method to obtain the solution of the collisionless
Boltzmann equation which describes the growth of large-scale structures as a
perturbative series over the initial density perturbations. We give an explicit
calculation of the second-order terms which are shown to agree with the results
obtained from the hydrodynamical description of the system. Then, we explain
that this identity extends to all orders of perturbation theory and that the
perturbative series actually diverge for hierarchical scenarios. However, since
the collisionless Boltzmann equation provides the exact description of the
dynamics (including the non-linear regime) these results may serve as a basis
for a study of the non-linear regime. In particular, we derive a
non-perturbative quadratic integral equation which explicitly relates the
actual non-linear distribution function to the initial conditions (more
precisely, to the linear growing mode). This allows us to write an explicit
path-integral expression for the probability distribution of the exact
non-linear density field.
