Enantiodromic effective generators of a Markov jump process with Gallavotti-Cohen symmetry

This paper deals with the properties of the stochastic generators of the effective (driven) processes associated with atypical values of transition-dependent time-integrated currents with Gallavotti-Cohen symmetry in Markov jump processes. Exploiting the concept of biased ensemble of trajectories by introducing a biasing field $s$, we show that the stochastic generators of the effective processes associated with the biasing fields $s$ and $E-s$ are enantiodromic with respect to each other where $E$ is the conjugated field to the current. We illustrate our findings by considering an exactly solvable creation-annihilation process of classical particles with nearest-neighbor interactions defined on a one-dimensional lattice.