  The evaluation of the path-integral representation for stochastic processes
in the weak-noise limit shows that these systems are governed by a set of
equations which are those of a classical dynamics. We show that, even when the
noise is colored, these may be put into a Hamiltonian form which leads to
better insights and improved numerical treatments. We concentrate on solving
Hamilton's equations over an infinite time interval, in order to determine the
leading order contribution to the mean escape time for a bistable potential.
The paths may be oscillatory and inherently unstable, in which case one must
use a multiple shooting numerical technique over a truncated time period in
order to calculate the infinite time optimal paths to a given accuracy. We look
at two systems in some detail: the underdamped Langevin equation driven by
external exponentially correlated noise and the overdamped Langevin equation
driven by external quasi-monochromatic noise. We deduce that caustics, focusing
and bifurcation of the optimal path are general features of all but the
simplest stochastic processes.
