  We consider a class of autonomous Hamiltonian systems subject to small,
time-periodic perturbations. When the perturbation parameter is set to zero,
the energy of the system is preserved. This is no longer the case when the
perturbation parameter is non-zero. We describe a topological method to
establish orbits which diffuse in energy for every suitably small perturbation
parameter $\varepsilon>0$. The method yields quantitative estimates: (i) The
existence of orbits along which the energy drifts by an amount independent of
$\varepsilon$. The time required by such orbits to drift is $O(1/\varepsilon)$;
(ii) The existence of orbits along which the energy makes chaotic excursions;
(iii) Explicit estimates for the Hausdorff dimension of the set of such chaotic
orbits; (iv) The existence of orbits along which the time evolution of energy
approaches a stopped diffusion process (Brownian motion with drift), as
$\varepsilon$ tends to $0$. For each $\varepsilon$ fixed, the set of initial
conditions of the orbits that yield the diffusion process has positive Lebesgue
measure, and in the limit the measure of these sets approaches zero. Moreover,
we can obtain any desired values of the drift and variance for the limiting
Brownian motion, for appropriate sets of initial conditions. A key feature of
our topological method is that it can be implemented in computer assisted
proofs. We give an application to a concrete model of the planar elliptic
restricted three-body problem, on the motion of an infinitesimal body relative
to the Neptune-Triton system.
