Statistical physics of long dynamical trajectories for a system in contact with several thermal reservoirs

For a system in contact with several reservoirs $r$ at different inverse-temperatures $\beta_r$, we describe how the Markov jump dynamics with the generalized detailed balance condition can be analyzed via a statistical physics approach of dynamical trajectories $[{\cal C}(t)]_{0 \leq t \leq T}$ over a long time interval $T \to + \infty$. The relevant intensive variables are the time-empirical density $\rho(\cal C)$, that measures the fractions of time spent in the various configurations ${\cal C}$, and the time-empirical jump densities $k_r ({\cal C', \cal C})$, that measure the frequencies of jumps from configuration ${\cal C}$ to configuration ${\cal C '}$ when it is the reservoir $r$ that furnishes or absorbs the corresponding energy difference ($E({\cal C '})- E({\cal C })$).