  We generalize the stochastic path integral formalism by considering
Hamiltonian dynamics in the presence of general Markovian noise. Kramers'
solution of the activation rate for escape over a barrier is generalized for
non-Gaussian driving noise in both the overdamped and underdamped limit. We
apply our general results to a Josephson junction detector measuring the
electron counting statistics of a mesoscopic conductor. Activation rate
dependence on the third current cumulant includes an additional term
originating from the back-action of the measurement circuit.
