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Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes

We study noisy heterogeneous diffusion processes with a position dependent diffusivity of the form $D(x)\sim D_0|x|^α$ in the presence of annealed and quenched disorder of the environment, corresponding to an effective variation of the exponent $α$ in time and space. In the case of annealed disorder, for which effectively $α=α(t)$ 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.