We are interested in the quasi-stationarity for the time-inhomogeneous Markov process
where $$ is a one-dimensional Brownian motion and κ ∈ (0, ∞). We first show that the law of X$$ conditioned not to go out from (−1, 1) until time t converges weakly towards the Dirac measure δ0 when $$, when t goes to infinity. Then, we show that this conditional probability measure converges weakly towards the quasi-stationary distribution for an Ornstein-Uhlenbeck process when $$. Finally, when $$, it is shown that the conditional probability measure converges towards the quasi-stationary distribution for a Brownian motion. We also prove the existence of a Q-process and a quasi-ergodic distribution for $$ and $$.
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DOI : 10.1051/ps/2020012
Mots-clés : Quasi-stationary distribution, $$-process, quasi-limiting distribution, quasi-ergodic distribution, Brownian motion
@article{PS_2020__24_1_661_0,
author = {Ocafrain, William},
title = {Quasi-stationarity for one-dimensional renormalized {Brownian} motion},
journal = {ESAIM: Probability and Statistics},
pages = {661--687},
publisher = {EDP-Sciences},
volume = {24},
year = {2020},
doi = {10.1051/ps/2020012},
mrnumber = {4170180},
zbl = {1454.60127},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2020012/}
}
TY - JOUR AU - Ocafrain, William TI - Quasi-stationarity for one-dimensional renormalized Brownian motion JO - ESAIM: Probability and Statistics PY - 2020 SP - 661 EP - 687 VL - 24 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2020012/ DO - 10.1051/ps/2020012 LA - en ID - PS_2020__24_1_661_0 ER -
Ocafrain, William. Quasi-stationarity for one-dimensional renormalized Brownian motion. ESAIM: Probability and Statistics, Tome 24 (2020), pp. 661-687. doi : 10.1051/ps/2020012. http://www.numdam.org/articles/10.1051/ps/2020012/
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