  We consider exit problems for small white noise perturbations of a dynamical
system generated by a vector field, and a domain containing a critical point
with all positive eigenvalues of linearization. We prove that, in the vanishing
noise limit, the probability of exit through a generic set on the boundary is
asymptotically polynomial in the noise strength, with exponent depending on the
mutual position of the set and the flag of the invariant manifolds associated
with the top eigenvalues. Furthermore, we compute the limiting exit
distributions conditioned on atypical exit events of polynomially small
probability and show that the limits are Radon--Nikodym equivalent to volume
measures on certain manifolds that we construct. This situation is in sharp
contrast with the large deviation picture where the limiting conditional
distributions are point masses.
