  Transition probabilities for stochastic systems can be expressed in terms of
a functional integral over paths taken by the system. Evaluating the integral
by the saddle point method in the weak-noise limit leads to a remarkable
mapping between dominant stochastic paths and conservative, Hamiltonian
mechanics in an effective potential. The conserved ``energy'' in the effective
system has dimensions of power. We show that this power, H, can be identified
with the Laplace parameter of the time-transformed dynamics. This
identification leads to insights into the non-equilibrium behaviour of the
system. Moreover, it facilitates the explicit summation over families of
trajectories, which is far harder in the time domain. This is crucial for
making contact with the long-time equilibrium limit.
