Quasi-stationarity for one-dimensional renormalized Brownian motion

Abstract

We are interested in the quasi-stationarity for the time-inhomogeneous Markov process$$X_t = \frac{B_t}{(t+1)^\kappa},$$where (Bt)t≥0is a one-dimensional Brownian motion andκ∈ (0,∞). We first show that the law ofXtconditioned not to go out from (−1, 1) until timetconverges weakly towards the Dirac measureδ0whenκ>½, whentgoes to infinity. Then, we show that this conditional probability measure converges weakly towards the quasi-stationary distribution for an Ornstein-Uhlenbeck process whenκ=½. Finally, whenκ<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 aQ-process and a quasi-ergodic distribution forκ=½ andκ<½.