  Transport in Hamiltonian systems with weak chaotic perturbations has been
much studied in the past. In this paper, we introduce a new class of problems:
transport in Hamiltonian systems with slowly changing phase space structure
that are not order one perturbations of a given Hamiltonian. This class of
problems is very important for many applications, for instance in celestial
mechanics. As an example, we study a class of one-dimensional Hamiltonians that
depend explicitly on time and on stochastic external parameters. The variations
of the external parameters are responsible for a distortion of the phase space
structures: chaotic, weakly chaotic and regular sets change with time. We show
that theoretical predictions of transport rates can be made in the limit where
the variations of the stochastic parameters are very slow compared to the
Hamiltonian dynamics. Exact asymptotic results can be obtained in the classical
case where the Hamiltonian dynamics is integrable for fixed values of the
parameters. For the more interesting chaotic Hamiltonian dynamics case, we show
that two mechanisms contribute to the transport. For some range of the
parameter variations, one mechanism -called transport by migration together
with the mixing regions - is dominant. We are then able to model transport in
phase space by a Markov model, the local diffusion model, and to give
reasonably good transport estimates.
