This paper establishes a process-level large deviations principle for Markov processes in the Euclidean space with a discontinuity in the transition mechanism along a hyperplane. The transition mechanism of the process is assumed to be continuous on one closed half-space and also continuous on the complementary open half-space. Similar results were recently obtained for discrete time processes by Dupuis and Ellis and by Nagot. Our proof relies on the work of Blinovskii and Dobrushin, which in turn is based on an earlier work of Dupuis and Ellis.
"On large deviations of Markov processes with discontinuous statistics." Ann. Appl. Probab. 8 (1) 45 - 66, February 1998. https://doi.org/10.1214/aoap/1027961033