On Laplace-Varadhan's integral lemma

In this Note we propose a complement to an integral lemma of Laplace-Varadhan arising in the literature of large deviations. We examine a situation in which the state space may depend on the rate of deviation. We have used this extension to analyze in a new way large deviation principles for the empirical measures on path space associated to interacting particle systems. We prove new large deviation principles on path space for an abstract class of discrete generation, as well as pure jump or McKean-Vlasov interacting particle systems.