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

In the weak noise limit ($D\to 0$), the escape rate over a one-dimensional potential barrier is $\Gamma \sim \exp (-\frac{S}{D})$. 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 $〈\xi \mathrm{}\left(t\right)\mathrm{}\xi \left(t^{\prime }\right)〉=\left(\frac{D}{\tau }\right)\exp \{-\frac{|t-t^{\prime }|}{\tau }\}$, the instanton satisfies a fourth-order, nonlinear, ordinary differential equation which can be readily solved numerically. Analytical results are derived for small and large $\tau$.