Uniform convergence to the $Q$-process

Abstract

The first aim of the present note is to quantify the speed of convergence of a conditioned process toward its Q-process under suitable assumptions on the quasi-stationary distribution of the process. Conversely, we prove that, if a conditioned process converges uniformly to a conservative Markov process which is itself ergodic, then it admits a unique quasi-stationary distribution and converges toward it exponentially fast, uniformly in its initial distribution. As an application, we provide a conditional ergodic theorem.

Q-processes and asymptotic properties of Markov processes conditioned not to hit moving boundaries

“…In this paper, the existence of a Q-process will be proved, as well as the exponential convergence in total variation of the probability measure P s,x (X [s,t] ∈ ·|τ A > T ) towards the Q-process, when T goes to infinity. In the same way as in the paper [8], this exponential convergence implies that the existence and the uniqueness of the quasi-ergodic distribution is equivalent to an ergodic theorem for the Q-process. In particular, this corollary will be applied for periodic moving boundaries to show the existence and the uniqueness of a quasi-ergodic distribution.…”

“…First we will show the exponential convergence towards the Qprocess essentially thanks to (18). In the second step, we will show the existence and uniqueness of the quasi-ergodic distribution using a method similar to that used in [8].…”

“…Then the authors show that (A1)-(A2) are equivalent to an exponential uniform convergence of the total variation distance between the conditional probability P µ (X t ∈ ·|τ A > t) and the unique quasi-stationary distribution. Moreover, one has, under these assumptions, the existence of a Q-process, as well as the existence and the uniqueness of the quasi-ergodic distribution (see [8] for this last result).…”

Quasi-stationarity for one-dimensional renormalized Brownian motion

“…Remark 5. In the time-homogeneous setting, it is usually expected that the quasi-ergodic distribution is the stationary distribution of the Q-process (see [5,8]). A similar result could even be expected in the time-inhomogeneous case when the Q-process converges weakly at the infinity (see [13]).…”

“…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.…”