Path integrals and non-Markov processes. III. Calculation of the escape-rate prefactor in the weak-noise limit

In earlier papers [McKane, Luckock, and Bray, Phys. Rev. A 41, 644 (1990); Bray, McKane, and Newman, ibid. 41, 657 (1990)] the path-integral approach was used to describe the behavior of a particle coupled to weak, exponentially correlated noise and moving in a one-dimensional potential. The method of steepest descents was used to calculate the leading-order exponential contributions to various quantities of physical interest. This analysis is developed further here. By accounting for small fluctuations about the paths of steepest descent, we determine the prefactors that multiply the dominant exponential contributions to these quantities. In particular, we calculate the escape rate for a particle over a potential barrier to second order in the noise correlation time (which is assumed to be short).