Quasi-stationarity for one-dimensional renormalized Brownian motion
ESAIM: Probability and Statistics, Tome 24 (2020), pp. 661-687.

We are interested in the quasi-stationarity for the time-inhomogeneous Markov process

X t =B t (t+1) κ ,

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
Classification : 60B10, 60F99, 60J50, 60J65
Mots-clés : Quasi-stationary distribution, $$-process, quasi-limiting distribution, quasi-ergodic distribution, Brownian motion 
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     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://archive.numdam.org/articles/10.1051/ps/2020012/}
}
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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://archive.numdam.org/articles/10.1051/ps/2020012/

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