  We study noisy heterogeneous diffusion processes with a position dependent
diffusivity of the form $D(x)\sim D_0|x|^\alpha$ in the presence of annealed
and quenched disorder of the environment, corresponding to an effective
variation of the exponent $\alpha$ in time and space. In the case of annealed
disorder, for which effectively $\alpha=\alpha(t)$ we show how the long time
scaling of the ensemble mean squared displacement (MSD) and the amplitude
variation of individual realizations of the time averaged MSD are affected by
the disorder strength. For the case of quenched disorder, the long time
behavior becomes effectively Brownian after a number of jumps between the
domains of a stratified medium. In the latter situation the averages are taken
over both an ensemble of particles and different realizations of the disorder.
As physical observables we analyze in detail the ensemble and time averaged
MSDs, the ergodicity breaking parameter, and higher order moments of the time
averages.
