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for FBM big scatter is not caused by higher dimensions in presence of reflecting walls. In fact, from Eqs. (4.2) and (4.6), we can analytically derive the relation 

which still holds in the case of confined motion (see ap- pendix B for the derivation). This relation is numerically confirmed in Fig. 7 where three E$_{B}$ curves collapse upon rescaling by dE$_{B}$ ( d ). According to this relation, we ex- pect that ergodic behavior obtained in one-dimensional confined motion (Figs. 3 and 4) will also be present in multiple dimensions with a factor of 1 /d . 

## V. FRACTIONAL LANGEVIN EQUATION MOTION IN CONFINED SPACE 

In this section we analyze FLE motion under confine- ment. Due to the different physical basis compared to FBM, in particular, the occurrence of inertia, we observe interesting variations on the properties studied in the pre- vious section. 

Using the correlation function [55] 

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## A. Mean squared displacement 

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one can show analytically that, similarly to the FBM, the ensemble averaged second moment 〈 y $^{2}$( t ) 〉 is identical to its time averaged analog 〈 δ $^{2}$(Δ) 〉 , for all Δ in free space, namely 

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Thus, the time averaged mean squared displacement turns over from a ballistic motion 

at short lag time to the subdiffusive behavior 

at long lag times, in free space. 

We numerically study how this scaling behavior is af- fected by the confinement. Figure 8 shows typical curves for the time averaged mean squared displacement, for in- terval sizes L = 1 / 2, 1, 3, and 100 (regarded as free space motion) with identical initial conditions and Hurst expo- nent H = 5 / 8. The results are summarized as follows: (1) We observe both scaling behaviors, 〈 δ $^{2}$(Δ , T ) 〉 ≃ Δ 2 

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