  We investigate the dynamics of a Bose-Einstein condensate in the presence of
a random potential created by optical speckles. We first consider the effect of
a weak disorder on the dipole and quadrupole collective oscillations, finding
uncorrelated frequency shifts of the two modes with respect to the pure
harmonic case. This behaviour, predicted by a sum rules approach, is confirmed
by the numerical solution of the Gross-Pitaevskii equation. Then we analyze the
role of disorder on the one-dimensional expansion in an optical guide,
discussing possible localization effects. Our theoretical analysis provides a
useful insight into the recent experiments performed at LENS [J. Lye et al.,
Phys. Rev. Lett. 95, 070401 (2005); C. Fort et al., cond-mat/0507144].
