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

TL;DR

This paper studies the long-term behavior of Markov processes conditioned to avoid moving boundaries, establishing criteria for convergence to Q-processes and distributions, with applications to diffusions coming down from infinity.

Contribution

It provides new criteria for exponential convergence to Q-processes and establishes the existence and uniqueness of quasi-ergodic and quasi-limit distributions for processes with moving or stabilizing boundaries.

Findings

Exponential convergence towards the Q-process established.

Existence and uniqueness of quasi-ergodic distributions proven.

Application to diffusions coming down from infinity.

Abstract

We investigate some asymptotic properties of general Markov processes conditioned not to be absorbed by 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 quasi-ergodic distribution considering either boundaries moving periodically or stabilizing boundaries. We also state the existence and uniqueness of quasi-limit distribution when absorbing boundaries stabilize. We finally deal with some examples such as diffusions which are coming down from infinity.