Analytical and numerical study of Kramers' exit problem I

Kramers' exit problem is concerned with noise activated escape from a potential well. In case the noise strength, ∈ is small this becomes a singular perturbation problem. The distribution of exit points on the separatrix (in the phase plane) is shown to be spread away from the saddle point, where the energy is minimal. The stochastic dynamics of the escaping trajectories, conditioned on not returning to a given critical energy contour, are studied analytically and numerically.