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.
"Uniform convergence to the $Q$-process." Electron. Commun. Probab. 22 1 - 7, 2017. https://doi.org/10.1214/17-ECP63