AbstractFor a class of Markov processes (in continuous or discrete time) we show that if the full large deviation holds for normalized occupation time measuresLt(w, ˙) with some rate functionJ, then the lower semicontinuous regularization ofJmust agree with the rate functionIintroduced by M. D. Donsker and S. R. S. Varadhan. As a consequence we show that for processes such as Brownian motion the full large deviation principle forLt(w, ˙) cannot hold withanyrate functio