We explore the properties of discrete-time stochastic processes with a bounded state space, whose
deterministic limit is given by a map of the unit interval. We nd that, in the mesoscopic description
of the system, the large jumps between successive iterates of the process can result in probability
leaking out of the unit interval, despite the fact that the noise is multiplicative and vanishes at the
boundaries. By including higher-order terms in the mesoscopic expansion, we are able to capture
the non-Gaussian nature of the noise distribution near the boundaries, but this does not preclude
the possibility of a trajectory leaving the interval. We propose a number of prescriptions for treating
these escape events, and we compare the results with those obtained for the metastable behavior of
the microscopic model, where escape events are not possible. We nd that, rather than truncating
the noise distribution, censoring this distribution to prevent escape events leads to results which
are more consistent with the microscopic model. The addition of higher moments to the noise
distribution does not increase the accuracy of the nal results, and it can be replaced by the simpler
Gaussian noise.