the ensemble averaged mean squared displacement. We also investigate a new quantity used to characterize the motion, the displacement correlation function. For these aspects we also study the dependence on the dimension- ality of the motion. The second aspect concerns how FBM and FLE mo- tions under confinement behave with respect to ergodic- ity. Experimentally the recording of single particle tra- jectories has become a standard tool, producing time se- ries of data that are then analyzed by time rather than ensemble averages. For subdiffusion processes both are not necessarily identical. In fact for CTRW subdiffusion with an ensemble averaged mean squared displacement of the form (1.1) the time averaged mean squared dis- placement The paper is organized as follows. In Sec. II, we in- troduce FBM and FLE motions, and review briefly their basic statistical properties. In Sec. III, we describe the numerical scheme for simulating FBM and FLE in con- fined space. Simulations results are presented in Sec. IV and V, where we discuss the effects of confinement and dimensionality on time-averaged mean squared displace- ment trajectory, ergodicity, and displacement correlation. We draw our Conclusions in Sec. VI. 2 on average scales like 〈 δ $^{2}$(Δ , T ) 〉 ≃ Δ /T 1 − α [23, 24]. That is, the anomalous diffusion is manifested only in the dependence on the overall measurement time T and not in the lag time Δ, that defines a window swept along the time series. Thus time and ensemble averages are indeed different. In contrast, for normal Brownian dif- fusion 〈 δ $^{2}$(Δ , T ) 〉 ≃ Δ is independent of T , and time and ensemble averages become identical, i.e., the sys- tem is ergodic. Different from CTRW subdiffusion, sys- tems governed by FBM or FLE are ergodic. The er- godicity breaking parameter measured from time aver- aged mean squared displacements converges algebraically to zero [ergodic behavior] as the measurement time in- creases, the convergence speed depending on the Hurst exponent H = α/ 2 [25]. For the FLE case, however, it was also shown that the ergodicity measured from the ve- locity variance can be broken for a class of colored noises [26]. One of the open questions in the context of ergodic- ity breaking for FBM and FLE in the above sense is the influence of boundary conditions on the time averages. It was shown for CTRW subdiffusion that confinement changes the short time scaling 〈 δ $^{2}$(Δ , T ) 〉 ≃ Δ /T 1 − α to the long time behavior 〈 δ $^{2}$(Δ , T ) 〉 ≃ (Δ /T ) 1 − α [27, 28]. Although we expect that FBM and FLE processes be- come stationary under confinement and, for instance, attain the same long-time mean squared displacement dictated by the size of the confinement volume, we in- vestigate how fast this relaxation actually is, and how it depends on the volume and the dimensionality. To this end we study the ergodicity breaking parameter for the system. We find that for both FBM and FLE con- finement actually decreases the value of the ergodicity breaking parameter with respect to unbounded motion, i.e., the process becomes more ergodic. Ergodicity is also enhanced with increasing dimensionality. We also discuss how FBM and FLE motions can be distinguished from time series from single particle trajectories. For easier comparison with other literature we introduced the Hurst exponent H that is related to the anomalous diffusion exponent via H = α/ 2. The Hurst exponent may vary in the range 0 < H < 1, such that FBM de- scribes both subdiffusion (0 < H < 1 / 2) and superdiffu- sion (1 / 2 < H < 1). The limits H = 1 / 2 and H = 1 cor- respond to Brownian and ballistic motion, respectively. with x $^{H}$(0) = 0; and the second moment scales like Eq. (1.1): FBM was originally introduced by Kolmogorov in 1940 [29] and further studied by Yaglom [30]. In a different context it was introduced by Mandelbrot in 1965 [31] and fully described by Mandelbrot and van Ness in 1968 in terms of a stochastic integral representation [32]. In the latter reference the authors wrote that ”We believe FBMs do provide useful models for a host of natural time series”. This study was motivated by Hurst’s analysis of annual river discharges [33], the observation that in eco- nomic time series cycles of all orders of magnitude oc- cur [34], and that many experimental studies exhibit the now famed 1 /f noise. FBM by now is widely used across fields. Among many others FBM has been identified as the underlying stochastic process of the subdiffusion of large molecules in biological cells [13, 35, 36]. We note that FBM is neither a semimartingale nor a Markov pro- cess, which makes it quite intricate to study with the tools of stochastic calculus [37, 38]. FBM, x $^{H}$( t ), is a Gaussian process with stationary in- crements which satisfies the following statistical proper- ties: the process is symmetric, We here define FBM and FLE. These two stochas- tic models share many common features, however, their physical nature is different. In the following we will see, in particular, how the two can be distinguished on the basis of experimental or simulations data. II. THEORETICAL MODEL A. Fractional Brownian motion δ $^{2}$(Δ , T ) = 1 T − Δ ∫ T − Δ 0 [ x ( t + Δ) − x ( t ) ] $^{2}$dt (1.2) 〈 x $^{H}$( t ) 〉 = 0 , (2.1) 〈 x $^{H}$( t ) $^{2}$〉 = 2 K$_{H}$t 2 $^{H}$. (2.2)