  We consider the thermally activated escape of an overdamped Brownian particle
over a potential barrier in the presence of periodic driving. A time-dependent
path-integral formalism is developed which allows us to derive asymptotically
exact weak-noise expressions for both the instantaneous and the time-averaged
escape rate. Our results comprise a conceptionally new, systematic treatment of
the rate prefactor multiplying the exponentially leading Arrhenius factor.
Moreover, an estimate for the deviations at finite noise-strengths is provided
and a supersymmetry-type property of the time averaged escape rate is verified.
For piecewise parabolic potentials, the rate-expression can be evaluated in
closed analytical form, while in more general cases, as exemplified by a cubic
potential, an action-integral remains to be minimized numerically. Our
comparison with very accurate numerical results demonstrates an excellent
agreement with the theoretical predictions over a wide range of driving
strengths and driving frequencies.
