  We propose a generalized diffusion equation for a flat Euclidean space
subjected to a continuous infinitesimal scale transform. For the special cases
of an algebraic or exponential expansion/contraction, governed by
time-dependent scale factors $a(t)\sim t^\lambda$ and $a(t)\sim \exp(\mu t)$,
the partial differential equation is solved analytically and the asymptotic
scaling behavior, as well as the dynamical exponents, are derived. Whereas in
the algebraic case the two processes (diffusion and expansion) compete and a
crossover is observed, we find that for exponential dynamics the expansion
dominates on all time scales. For the case of contracting spaces, an algebraic
evolution slows down the overall dynamics, reflected in terms of a new
effective diffusion constant, whereas an exponential contraction neutralizes
the diffusive behavior entirely and leads to a stationary state. Furthermore,
we derive various first-passage properties and describe four qualitatively
different regimes of (strong) recurrent/transient behavior depending on the
scale factor exponent.
