  We demonstrate the non-ergodicity of a simple Markovian stochastic processes
with space-dependent diffusion coefficient $D(x)$. For power-law forms $D(x)
\simeq|x|^{\alpha}$, this process yield anomalous diffusion of the form $\ <
x^2(t)\ > \simeq t^{2/(2-\alpha)}$. Interestingly, in both the sub- and
superdiffusive regimes we observe weak ergodicity breaking: the scaling of the
time averaged mean squared displacement $\{\delta^2}$ remains \emph{linear} and
thus differs from the corresponding ensemble average $\ <x^2(t)\ >$. We analyze
the non-ergodic behavior of this process in terms of the ergodicity breaking
parameters and the distribution of amplitude scatter of $\{\delta^2}$. This
model represents an alternative approach to non-ergodic, anomalous diffusion
that might be particularly relevant for diffusion in heterogeneous media.
