Quasi-stationarity for one-dimensional renormalized Brownian motion
Résumé
We are interested in the quasi-stationarity of the time-inhomogeneous Markov process X t = B t (t + 1) κ where (B t) t≥0 is a one-dimensional Brownian motion and κ ∈ (0, ∞). 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 δ 0 when κ > 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 κ = 1 2. Finally, when κ < 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 κ = 1 2 and κ < 1 2 .
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