  We consider the fluctuations of a time-integrated particle current around an
atypical value in a generic stochastic Markov process involving classical
particles with two-site interaction and hardcore repulsion on a finite
one-dimensional lattice with open boundaries. We address the question of which
interactions one has to impose on such process to make the atypical value of
the current typical. It is known that a corresponding effective stochastic
Markov process might exist whose typical value of the current is equal to the
atypical value of the current in the original process within a
time-translational invariant regime. This effective process has, in principle,
non-local transition rates. Nevertheless, it turns out that under some
conditions the stochastic generator of the effective process has the same
dynamical rules as the stochastic generator of the original process. We find
these conditions and show that our approach can be generalized to any
time-integrated observable.
