The problem of finding a necessary and sufficient condition for the continuity of the local times for a general Markov process is still open. Barlow and Hawkes have completely treated the case of the Lévy processes, and Marcus and Rosen have solved the case of the strongly symmetric Markov processes. We treat here the continuity of the local times of Borel right processes. Our approach unifies that of Barlow and Hawkes and of Marcus and Rosen, by using an associated Gaussian process, that appears as a limit in a CLT involving the local time process.
"On the continuity of local times of Borel right Markov processes." Ann. Probab. 35 (3) 915 - 934, May 2007. https://doi.org/10.1214/009117906000000980