  We consider a continuous random walk model for describing normal as well as
anomalous diffusion of particles subjected to an external force when these
particles diffuse in a uniformly expanding (or contracting) medium. A general
equation that relates the probability distribution function (pdf) of finding a
particle at a given position and time to the single-step jump length and
waiting time pdfs is provided. The equation takes the form of a generalized
Fokker-Planck equation when the jump length pdf of the particle has a finite
variance. This generalized equation becomes a fractional Fokker-Planck equation
in the case of a heavy-tailed waiting time pdf. These equations allow us to
study the relationship between expansion, diffusion and external force. We
establish the conditions under which the dominant contribution to transport
stems from the diffusive transport rather than from the drift due to the medium
expansion. We find that anomalous diffusion processes under a constant external
force in an expanding medium described by means of our continuous random walk
model are not Galilei invariant, violate the generalized Einstein relation, and
lead to propagators that are qualitatively different from the ones found in a
static medium. Our results are supported by numerical simulations.
