Some notes on large deviations of Markov processes

In this paper pie shall characterize the large deviation principles (abbrev iated to LDP) of Donsker-Varadhan of a Markov process both for the weak con vergence topology and for the tau-topology, by means of a hyper-exponential recurrence property. A Lyapunov criterion for this type of recurrence prop erty is presented. These results are applied to countable Markov chains, un idimensional diffusions, elliptic or hypoelliptic diffusions on Riemannian manifolds. Several counter-examples are equally presented.