Kramers' law for a bistable system with time-delayed noise

We demonstrate that the classical Kramers' escape problem can be extended to describe a bistable system under the influence of noise consisting of the superposition of a white Gaussian noise with the same noise delayed by time τ. The distribution of times between two consecutive switches decays piecewise exponentially, and the switching rates for 0<t<τ and τ<t<2τ are calculated analytically using the Langevin equation. These rates are different since, for the particles remaining in one well for longer than τ, the delayed noise acquires a nonzero mean value and becomes negatively autocorrelated. To account for these effects we define an effective potential and an effective diffusion coefficient of the delayed noise.