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We investigate some asymptotic properties of general Markov processesconditioned not to be absorbed by moving boundaries. We first give generalcriteria involving an exponential convergence towards the Q-process, that isthe law of the considered Markov process conditioned never to reach the movingboundaries. This exponential convergence allows us to state the existence anduniqueness of quasi-ergodic distribution considering either boundaries movingperiodically or stabilizing boundaries. We also state the existence anduniqueness of quasi-limit distribution when absorbing boundaries stabilize. Wefinally deal with some examples such as diffusions which are coming down frominfinity.

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