Quasi-stationarity for one-dimensional renormalized Brownian motion

International audienceWe are interested in the quasi-stationarity for the time-inhomogeneous Markov process$$X_t = \frac{B_t}{(t+1)^\kappa},$$where (Bt)t≥0 is a one-dimensional Brownian motion and κ ∈ (0, ∞). We first show that the law of Xt conditioned not to go out from (−1, 1) until time t converges weakly towards the Dirac measure δ0 when κ>½, when t goes to infinity. Then, we show that this conditional probability measure converges weakly towards the quasi-stationary distribution for an Ornstein-Uhlenbeck process when κ=½. Finally, when κ<½, it is shown that the conditional probability measure converges towards the quasi-stationary distribution for a Brownian motion. We also prove the existence of a Q-process and a quasi-ergodic distribution for κ=½ and κ