  Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of
passive tracers in complex and biological systems. It is a highly
non-stationary process governed by the Langevin equation for Brownian motion,
however, with a power-law time dependence of the noise strength. Here we study
the aging properties of SBM for both unconfined and confined motion.
Specifically, we derive the ensemble and time averaged mean squared
displacements and analyze their behavior in the regimes of weak, intermediate,
and strong aging. A very rich behavior is revealed for confined aging SBM
depending on different aging times and whether the process is sub- or
superdiffusive. We demonstrate that the information on the aging factorizes
with respect to the lag time and exhibits a functional form, that is identical
to the aging behavior of scale free continuous time random walk processes.
While SBM exhibits a disparity between ensemble and time averaged observables
and is thus weakly non-ergodic, strong aging is shown to effect a convergence
of the ensemble and time averaged mean squared displacement. Finally, we derive
the density of first passage times in the semi-infinite domain that features a
crossover defined by the aging time.
