  We consider the overdamped limit of two-dimensional double well systems
perturbed by weak noise. In the weak noise limit the most probable
fluctuational path leading from either point attractor to the separatrix (the
most probable escape path, or MPEP) must terminate on the saddle between the
two wells. However, as the parameters of a symmetric double well system are
varied, a unique MPEP may bifurcate into two equally likely MPEP's. At the
bifurcation point in parameter space, the activation kinetics of the system
become non-Arrhenius. In this paper we quantify the non-Arrhenius behavior of a
system at the bifurcation point, by using the Maslov-WKB method to construct an
approximation to the quasistationary probability distribution of the system
that is valid in a boundary layer near the separatrix. The approximation is a
formal asymptotic solution of the Smoluchowski equation. Our analysis relies on
the development of a new scaling theory, which yields `critical exponents'
describing weak-noise behavior near the saddle, at the bifurcation point.
