In this note we show a simple formula for the joint density of local times, last exit tree and cycling numbers of continuous-time Markov chains on finite graphs, which involves the modified Bessel function of the first type.
"Explicit formula for the density of local times of Markov jump processes." Electron. Commun. Probab. 23 1 - 7, 2018. https://doi.org/10.1214/18-ECP194