  We study small white noise perturbations of planar dynamical systems with
heteroclinic networks in the limit of vanishing noise. We show that the
probabilities of transitions between various cells that the network tessellates
the plane into decay as powers of the noise magnitude. We show that the most
likely scenario for the realization of these rare transition events involves
spending atypically long times in the neighborhoods of certain saddle points of
the network. We describe the hierarchy of time scales and clusters of
accessibility associated with these rare transition events. We discuss
applications of our results to homogenization problems and to the invariant
distribution asymptotics. At the core of our results are local limit theorems
for exit distributions obtained via methods of Malliavin calculus.
