Bistability driven by weakly colored Gaussian noise: The Fokker-Planck boundary layer and mean first-passage times

We develop a singular perturbation approach to the problem of first-passage times for non-Markovian processes driven by Gaussian colored noise near the white-noise limit. In particular we treat the problem of overdamped tunneling in a bistable quartic potential in the presence of Gaussian noise of short correlation time τ. The correct treatment of the absorbing boundary yields a lowest-order correction proportional to ${\mathrm{\tau }}^{1/2}$ with a proportionality constant involving the Milne extrapolation length for the Fokker-Planck equation, given in terms of the Riemann ζ function.