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| - Quasi-stationarity for one-dimensional renormalized Brownian motion
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| 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≥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.
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| - © The authors. Published by EDP Sciences, SMAI 2020
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| - The authors. Published by EDP Sciences, SMAI
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