Quasi-stationarity for one-dimensional renormalized Brownian motion

Abstract

We are interested in the quasi-stationarity of the time-inhomogeneous Markov processIn the literature, this notion is also called mean-ratio quasi-stationary distribution. The references [10,12] does not deal with quasi-ergodic distribution. See for example [5,8] which provide general assumptions implying the existence of quasi-ergodic distribution for time-homogeneous Markov processes.Some general results on quasi-stationarity for time-inhomogeneous Markov process are established, particularly in [9], 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. However it will be difficult to apply these results for our purpose. See also [16,11,14,13] for a few