Rs. Maier et Dl. Stein, A SCALING THEORY OF BIFURCATIONS IN THE SYMMETRICAL WEAK-NOISE ESCAPEPROBLEM, Journal of statistical physics, 83(3-4), 1996, pp. 291-357
We consider two-dimensional overdamped double-well systems perturbed b
y white 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 th
e two wells. However, as the parameters of a symmetric double-well sys
tem are varied, a unique MPEP may bifurcate into two equally likely MP
EPs. At the bifurcation point in parameter space, the activation kinet
ics of the system become non-Arrhenius. 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 probabili
ty distribution of the system that is valid in a boundary layer near t
he separatrix. The approximation is a Formal asymptotic solution of th
e Smoluchowski equation. Our construction relies on a new scaling theo
ry, which yields ''critical exponents'' describing weak-noise behavior
at the bifurcation point, near the saddle.