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

Abstract

We investigate some asymptotic properties of general Markov processes conditioned not to be absorbed by the moving boundaries. We first give general criteria involving an exponential convergence towards the Q-process, that is the law of the considered Markov process conditioned never to reach the moving boundaries. This exponential convergence allows us to state the existence and uniqueness of the quasiergodic distribution considering either boundaries moving periodically or stabilizing boundaries. We also state the existence and uniqueness of a quasi-limiting distribution when absorbing boundaries stabilize. We finally deal with some examples such as diffusions which are coming down from infinity.

Quasi-stationarity for one-dimensional renormalized Brownian motion

“…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]). It is therefore astonishing to see that this is not the case for our process in the critical regime, even though the Q-process admits a stationary measure.…”

“…• This is straightforward using the Harnack inequality for a Brownian motion and using the change of time provided by the Dubin-Schwartz's transformation (13). Now let us state and prove Lemma 2.…”

“…However it will be difficult to apply these results for our purpose. See also [16,11,14,13] for a few…”

“…In this paper, we are interested in some notions related to quasi-stationarity for a one-dimensional Brownian motion (B t ) t≥0 killed when crossing the moving boundaries t → (−(t + 1) κ , (t + 1) κ ), with κ ∈ (0, ∞). Quasistationarity with moving boundaries was studied in [13] and [14] for periodic or converging boundaries, but expanding boundaries were not yet considered. Actually, instead of considering the process B absorbed at t → (−(t + 1) κ , (t + 1) κ ), we will study the quasi-stationarity of the process X = (X t ) t≥0 absorbed at (−1, 1) c and defined by…”

An ergodic theorem for asymptotically periodic time-inhomogeneous Markov processes, with application to quasi-stationarity with moving boundaries

“…One point of interest in this paper is to apply the main result (3) to the theory of quasi-stationarity with moving boundaries, which refers to the study of asymptotical behaviors for Markov processes conditioned not to reach a moving subset of the state space. See in particular [24,25], where a "conditional ergodic theorem" (see further the definition of a quasi-ergodic distribution) has been shown when the absorbing boundaries move periodically. In this paper, one shows that a similar result holds when the boundary is asymptotically periodic, assuming that the process satisfies a conditional Doeblin's condition (see Assumption (A')).…”