The explicit form of the rate function for semi-Markov processes and its contractions

We derive the explicit form of the rate function for semi-Markov processes. Here, the “random time change trick” plays an essential role. Also, by exploiting the contraction principle of the large deviation theory to the explicit form, we show that the fluctuation theorem (Gallavotti-Cohen Symmetry) holds for semi-Markov cases. Furthermore, we elucidate that our rate function is an extension of the Level 2.5 rate function for Markov processes to semi-Markov cases.