  We study the first passage dynamics of an ageing stochastic process in the
continuous time random walk (CTRW) framework. In such CTRW processes the test
particle performs a random walk, in which successive steps are separated by
random waiting times distributed in terms of the waiting time probability
density function $\psi(t) \simeq t^{-1-\alpha}$) ($0\le \alpha \le 2$). An
ageing stochastic process is defined by the explicit dependence of its dynamic
quantities on the ageing time $t_a$, the time elapsed between its preparation
and the start of the observation. Subdiffusive ageing CTRWs describe systems
such as charge carriers in amorphous semiconductors, tracer dispersion in
geological and biological systems, or the dynamics of blinking quantum dots. We
derive the exact forms of the first passage time density for an ageing
subdiffusive CTRW in the semi-infinite, confined, and biased case, finding
different scaling regimes for weakly, intermediately, and strongly aged
systems: these regimes, with different scaling laws, are also found when the
scaling exponent is in the range $1<\alpha<2$, for sufficiently long $t_a$. We
compare our results with the ageing motion of a test particle in a quenched
energy landscape. We test our theoretical results in the quenched landscape
against simulations: only when the bias is strong enough, the correlations from
returning to previously visited sites become insignificant and the results
approach the aging CTRW results. With small or without bias, the ageing effects
disappear and a change in the exponent compared to the case of a completely
annealed landscape can be found, reflecting the build-up of correlations in the
quenched landscape.
