  It is known that the stochastic generators of effective processes associated
with the unconditioned dynamics of rare events might consist of non-local
interactions; however, it can be shown that there are special cases for which
these generators can include local interactions. In this paper we investigate
this possibility by considering systems of classical particles moving on a
one-dimensional lattice with open boundaries. The particles might have
hard-core interactions similar to the particles in an exclusion process or
there can be arbitrary many particles at single site as in a zero-range
process. Assuming that the interactions in the original process are local and
site-independent, we will show that under certain constraints on the
microscopic reaction rules, the stochastic generator of unconditioned process
can be local but site-dependent. As two examples, the asymmetric
zero-temperature Glauber model and the A-model with diffusion are presented and
studied under the above mentioned constraints.
