  Surface diffusion of small adsorbates is analyzed in terms of the so-called
intermediate scattering function and dynamic structure factor, observables in
experiments using the well-known quasielastic Helium atom scattering and Helium
spin echo techniques. The linear theory used is an extension of the neutron
scattering due to van Hove and considers the time evolution of the position of
the adsorbates in the surface. This approach allows us to use a stochastic
trajectory description following the classical, quantum and Bohmian frameworks.
Three regimes of motion are clearly identified in the diffusion process:
ballistic, Brownian and intermediate which are well characterized, for the
first two regimes, through the mean square displacements and Einstein relation
for the diffusion constant. The Langevin formalism is used by considering Ohmic
friction, moderate surface temperatures and small coverages. In the Bohmian
framework, the starting point is the so-called Schr\"odinger-Langevin equation
which is a nonlinear, logarithmic differential equation. By assuming a Gaussian
function for the probability density, the corresponding quantum stochastic
trajectories are given by a dressing scheme consisting of a classical
stochastic trajectory of the center of the Gaussian wave packet, issued from
solving the Langevin equation (particle property), plus the time evolution of
its width governed by the damped Pinney differential equation (wave property).
The velocity autocorrelation function is the same as the classical one when the
initial spread rate is assumed to be zero. If not, in the diffusion regime, the
Brownian-Bohmian motion shows a weak anomalous diffusion.
