  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.
