  We present a classical, mesoscopic derivation of the Fokker-Planck equation
for diffusion in an expanding medium. To this end, we take a conveniently
generalized Chapman-Kolmogorov equation as the starting point. We obtain an
analytical expression for the Green's function (propagator) and investigate
both analytically and numerically how this function and the associated moments
behave. We also study first-passage properties in expanding hyperspherical
geometries. We show that in all cases the behavior is determined to a great
extent by the so-called Brownian conformal time $\tau(t)$, which we define via
the relation $\dot \tau=1/a^2$, where $a(t)$ is the expansion scale factor. If
the medium expansion is driven by a power law [$a(t) \propto t^\gamma$ with
$\gamma>0$], we find interesting crossover effects in the mixing effectiveness
of the diffusion process when the characteristic exponent $\gamma$ is varied.
Crossover effects are also found at the level of the survival probability and
of the moments of the first passage-time distribution with two different
regimes separated by the critical value $\gamma=1/2$. The case of an
exponential scale factor is analyzed separately both for expanding and
contracting media. In the latter situation, a stationary probability
distribution arises in the long time limit.
