  The exit problem for small perturbations of a dynamical system in a domain is
considered. It is assumed that the unperturbed dynamical system and the domain
satisfy the Levinson conditions. We assume that the random perturbation affects
the driving vector field and the initial condition, and each of the components
of the perturbation follows a scaling limit. We derive the joint scaling limit
for the random exit time and exit point. We use this result to study the
asymptotics of the exit time for 1-d diffusions conditioned on rare events.
