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We consider the fluctuations of a time-integrated particle current around anatypical value in a generic stochastic Markov process involving classicalparticles with two-site interaction and hardcore repulsion on a finiteone-dimensional lattice with open boundaries. We address the question of whichinteractions one has to impose on such process to make the atypical value ofthe current typical. It is known that a corresponding effective stochasticMarkov process might exist whose typical value of the current is equal to theatypical value of the current in the original process within atime-translational invariant regime. This effective process has, in principle,non-local transition rates. Nevertheless, it turns out that under someconditions the stochastic generator of the effective process has the samedynamical rules as the stochastic generator of the original process. We findthese conditions and show that our approach can be generalized to anytime-integrated observable.

On Effective Stochastic Generators for Conditioned Dynamics at an Atypical Reaction-Diffusion Current