  Self-similar dynamical processes are characterized by a growing length scale
$\xi$ which increases with time as $\xi \sim t^{1/z}$, where z is the dynamical
exponent. The best known example is a simple random walk with z=2. Usually such
processes are assumed to take place on a static background. In this paper we
address the question what changes if the background itself evolves dynamically.
As an example we consider a random walk on an isotropically and homogeneously
inflating space. For an exponentially fast expansion it turns out that the
self-similar properties of the random walk are destroyed. For an inflation with
power-law characteristics, however, self-similarity is preserved provided that
the exponent controlling the growth is small enough. The resulting probability
distribution is analyzed in terms of cumulant ratios. Moreover, the dynamical
exponent z is found to change continuously with the control exponent.
