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Transient ageing in fractional Brownian and Langevin equation motion

Stochastic processes driven by stationary fractional Gaussian noise, that is, fractional Brownian motion and fractional Langevin equation motion, are usually considered to be ergodic in the sense that, after an algebraic relaxation, time and ensemble averages of physical observables coincide. Recently it was demonstrated that fractional Brownian motion and fractional Langevin equation motion under external confinement are transiently non-ergodic---time and ensemble averages behave differently---from the moment when the particle starts to sense the confinement. Here we show that these processes also exhibit transient ageing, that is, physical observables such as the time averaged mean squared displacement depend on the time lag between the initiation of the system at time $t=0$ and the start of the measurement at the ageing time $t_a$. In particular, it turns out that for fractional Langevin equation motion the ageing dependence on $t_a$ is different between the cases of free and confined motion. We obtain explicit analytical expressions for the aged moments of the particle position as well as the time averaged mean squared displacement and present a numerical analysis of this transient ageing phenomenon.