  Motivated by subdiffusive motion of bio-molecules observed in living cells we
study the stochastic properties of a non-Brownian particle whose motion is
governed by either fractional Brownian motion or the fractional Langevin
equation and restricted to a finite domain. We investigate by analytic
calculations and simulations how time-averaged observables (e.g., the time
averaged mean squared displacement and displacement correlation) are affected
by spatial confinement and dimensionality. In particular we study the degree of
weak ergodicity breaking and scatter between different single trajectories for
this confined motion in the subdiffusive domain. The general trend is that
deviations from ergodicity are decreased with decreasing size of the movement
volume, and with increasing dimensionality. We define the displacement
correlation function and find that this quantity shows distinct features for
fractional Brownian motion, fractional Langevin equation, and continuous time
subdiffusion, such that it appears an efficient measure to distinguish these
different processes based on single particle trajectory data.
