A strong large deviation principle for the empirical measure of random walks

In this article we show that the empirical measure of certain continuous time random walks satisfies a strong large deviation principle with respect to a topology introduced in~{MV2016} by Mukherjee and Varadhan. This topology is natural in models which exhibit an invariance with respect to spatial translations. Our result applies in particular to the case of simple random walk and complements the results obtained in~{MV2016} in which the large deviation principle has been established for the empirical measure of Brownian motion.