Transient solution of the Kramers problem in the weak noise limit

The one-dimensional escape problem for both overdamped and underdamped cases is treated using a combination of matched asymptotic and Laplace transformation techniques. It is shown that the shape of transient curves for the probability flux at the top of the barrier is insensitive to the specific shape of the potential, but is determined only by the ratio of the characteristic time scales at the points of stable and unstable equilibria. For the overdamped case this results in a relatively small number of possible types of transient behavior for various potentials (examples of quartic, slanted sinusoidal, cubic, and nonanalytic potentials are considered). In the underdamped situation the transient curve is identical for any shape of potential.