Instanton Calculation of the Escape Rate for Activation over a Potential Barrier Driven by Colored Noise

In the weak noise limit (D→0), the escape rate over a one-dimensional potential barrier is Γ~exp(-S/D). Here S is calculated exactly by identifying appropriate instanton solutions in a path-integral formulation of the Langevin equation. For the "colored noise" problem, defined by the noise correlator 〈ξ(t)ξ(t^')〉=(Dτ)exp-|t-t^'|τ, the instanton satisfies a fourth-order, nonlinear, ordinary differential equation which can be readily solved numerically. Analytical results are derived for small and large τ.