content_image/420378.jpg content_image/420380.jpg content_image/420379.jpg content_image/420376.jpg
<img src='content_image/420378.jpg'>

<img src='content_image/420380.jpg'>

We present the time averaged displacement correlation for confined subdiffusive motion ( H = 1 / 4) in Fig. 5. Be- cause of the negativity of expression (4.5) the absolute value of the displacement correlation is drawn in the log- log representation. At short lag times the slope of the correlation functions is proportional to 2 H as expected 

<img src='content_image/420379.jpg'>

from Eq. (4.5). However, at long lag times, we inter- estingly observe fluctuations of the correlations around a constant value, reflecting the confinement of the motion. 

To mimic the anomalous diffusion of particles in- side biological cells, we also simulate two- and three- dimensional FBM based on Eq. (2.7) in the presence of reflecting walls. In free space, the ensemble average of the time averaged mean squared displacement is simply given by 

<img src='content_image/420376.jpg'>

i.e., it is additive as for the ensemble average. This be- havior is indeed observed in Fig. 6 where five different mean squared displacement curves are drawn for L = 1 in 1D, 2D, and 3D, respectively. Only the height of the curves are affected by the dimensionality. There is no noticeable difference in the scatter of the curves. 

We further investigate the effects of dimensionality on the scatter of the mean squared displacement curves, as possibly the strong scatter observed in experiments [8– 10] may also occur for FBM in higher dimensions. To see the effect of dimensionality on the ergodicity behavior we measure E$_{B}$ versus lag time for one-, two-, and three- dimensional embedding dimension for the same values of L and H . Interestingly, the result shows that E$_{B}$ tends to decrease with increasing dimensionality d , meaning that 

## D. Dimensionality 

