Quasi-stationarity for one-dimensional renormalized Brownian motion

Institut de Mathématiques de Toulouse, UMR 5219; Université de Toulouse, CNRS, UPS IMT, 31062 Toulouse Cedex 9, France

^* Corresponding author: w.ocafrain@hotmail.fr

Received: 19 July 2019
Accepted: 8 March 2020

Abstract

We are interested in the quasi-stationarity for the time-inhomogeneous Markov process $$X_t=\frac{B_t}{{\left(t+1\right)}^{\kappa }},$$

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 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.