  We study the stochastic dynamics of a particle in a periodically driven
potential. For atomic ions trapped in radio-frequency Paul traps, noise heating
and laser cooling typically act slowly in comparison with the unperturbed
motion. These stochastic processes can be accounted for in terms of a
probability distribution defined over the action variables, which would
otherwise be conserved within the regular regions of the Hamiltonian phase
space. We present a semiclassical theory of low-saturation laser cooling
applicable from the limit of low-amplitude motion to large-amplitude motion,
accounting fully for the time-dependent and anharmonic trap. We employ our
approach to a detailed study of the stochastic dynamics of a single ion,
drawing general conclusions regarding the nonequilibrium dynamics of
laser-cooled trapped ions. We predict a regime of anharmonic motion in which
laser cooling becomes diffusive (i.e., it is equally likely to cool the ion as
it is to heat it), and can also turn into effective heating. This implies that
a high-energy ion could be easily lost from the trap despite being laser
cooled; however, we find that this loss can be counteracted using a laser
detuning much larger than Doppler detuning.
