Large deviations for non-irreducible Markov chains on Euclidean spaces

We establish the weak large deviations principle for empirical measures of Markov chains on $\mathbb R^d$ under mild assumptions. In particular, no irreducibility is assumed and the initial measure may be arbitrary. The proof is entirely self-contained and relies on subadditivity. In the absence of irreducibility, examples show that the rate function is not convex in general.