Volume 24, 2020
|Page(s)||661 - 687|
|Published online||04 November 2020|
Quasi-stationarity for one-dimensional renormalized Brownian motion
Institut de Mathématiques de Toulouse, UMR 5219; Université de Toulouse, CNRS, UPS IMT,
Toulouse Cedex 9, France
* Corresponding author: firstname.lastname@example.org
Accepted: 8 March 2020
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<½.
Mathematics Subject Classification: 60B10 / 60F99 / 60J50 / 60J65
Key words: Quasi-stationary distribution / Q-process / quasi-limiting distribution / quasi-ergodic distribution / Brownian motion
© The authors. Published by EDP Sciences, SMAI 2020
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