  We are interested in the quasi-stationarity of the time-inhomogeneous Markov
process X t = B t (t + 1) $\kappa$ where (B t) t$\ge$0 is a one-dimensional
Brownian motion and $\kappa$ $\in$ (0, $\infty$). We first show that the law of
X t conditioned not to go out from (--1, 1) until the time t converges weakly
towards the Dirac measure $\delta$ 0 when $\kappa$ > 1 2 as t goes to infinity.
Then we show that this conditioned probability converges weakly towards the
quasi-stationary distribution of an Ornstein-Uhlenbeck process when $\kappa$ =
1 2. Finally, when $\kappa$ < 1 2 , it is shown that the conditioned
probability converges towards the quasi-stationary distribution of a Brownian
motion. We also prove the existence of a Q-process and a quasi-ergodic
distribution for $\kappa$ = 1 2 and $\kappa$ < 1 2 .
