Quasi-stationarity for one-dimensional renormalized Brownian motion

William Ocafrain

doi:10.1051/ps/2020012

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.

Highlights

In this paper, we are interested in some notions related to quasi-stationarity for a one-dimensional Brownian motion (Bt)t≥0 killed when reaching the moving boundary t → {−(t + 1)κ, (t + 1)κ}, with κ ∈ (0, ∞)
We prove the existence of a Q-process and a quasi-ergodic distribution for κ=
We are interested in some notions related to quasi-stationarity for a one-dimensional Brownian motion (Bt)t≥0 killed when reaching the moving boundary t → {−(t + 1)κ, (t + 1)κ}, with κ ∈ (0, ∞)

Summary

Introduction

We are interested in some notions related to quasi-stationarity for a one-dimensional Brownian motion (Bt)t≥0 killed when reaching the moving boundary t → {−(t + 1)κ, (t + 1)κ}, with κ ∈ (0, ∞). See for example [5, 9] which provide general assumptions implying the existence of quasi-ergodic distributions for time-homogeneous Markov processes. It is shown in [5] that, if the Q-process is Harris recurrent, the quasi-ergodic distribution is the stationary distribution of the Q-process. Concerning the time-inhomogeneous setting, similar results can be stated (see [16]) when the Q-process converges weakly at infinity In this case, the quasi-ergodic distribution coincides with the limiting probability measure. Some general results on quasi-stationarity for time-inhomogeneous Markov process are established, in [10], where it is shown that criteria based on Doeblin-type condition implies a mixing property (or merging or weak ergodicity) and the existence of the Q-process.

X is is supercritical critical

Existence and uniqueness of a quasi-stationary distribution

Quasi-stationarity for Y

Quasi-ergodic distribution