Space-time large deviations for interacting particle systems

We prove a process-level large deviation principle for the space-time empirical averages of continuous-time systems on an infinite lattice. Our methods rely on the Donsker-Varadhan large deviation theory for Markov processes, and allow us to express the rate function rather explicitly in terms of the Markov generator of the infinite particle system. We can prove our principle for a large class of spin systems with no particle exchange, as well as for infinite diffusion processes whose drift is the gradient of a finite range Hamiltonian.