Drift and diffusion in phase space

The problem of stability of the action variables (\ie of the adiabatic invariants) in perturbations of completely integrable (real analytic) hamiltonian systems with more than two degrees of freedom is considered. Extending the analysis of {\rm [A]}, we work out a general quantitative theory, from the point of view of {\sl dimensional analysis}, for {\sl a priori unstable systems} (\ie systems for which the unperturbed integrable part possesses separatrices), proving, in general, the existence of the so--called Arnold's diffusion and establishing upper bounds on the time needed for the perturbed action variables to {\sl drift} by an amount of $O(1)$.