Analytical and numerical study of Kramers' exit problem II

Kramers' exit problem is concerned with noise activated escape from a potential well. In case the noise strength, ε, (temperature measured in units of potential barrier height) is small this becomes a singular perturbation problem. It is shown that actually, most of the probability of the exit points on the separatrix is located at a distance O(ε) from the saddle point and the probability vanishes altogether at the saddle point. The stochastic dynamics of the escaping trajectories, conditioned on not returning to a given critical energy contour, are studied analytically and numerically.