Large deviations for surviving trajectories of general Markov processes

The purpose of this paper is to ensure the conditions of G\"artner-Ellis Theorem for evaluations of the empirical measure. We show that up-to-date conditions for ensuring the convergence to a quasi-stationary distribution can be applied efficiently. By this mean, we are able to prove Large Deviation results even with a conditioning that the process is not extinct at the end of the evaluation. The domain on which these Large Deviation results apply is implicitly given by the range of penalization for which one can prove the above-mentioned results of quasi-stationarity. We propose a way to relate the range of controlled deviations to the range of admissible penalization. Central Limit Theorems are deduced from these results of Large Deviations. As an application, we consider the empirical measure of position and jumps of a continuous-time process on aunbounded domain of $R^d$. This model is inspired by the adaptation of a population to a changing environment. Jumps makes it possible for the process to face a deterministic dynamics leading to high extinction areas.
Comment: The argument in the proof of Proposition 1.3.4 is wrong and I cannot give simple conditions ensuring that the function is strictly increasing. This makes the statements much less clear if we cannot ensure it, so I hope I will be able to handle it. Besides, the limit in Lemma 3.3.2 is wrong. Nonetheless, the proof that the variance goes to 0 is true