Path integrals and non-Markov processes. II. Escape rates and stationary distributions in the weak-noise limit

The path-integral formalism developed in the preceding paper [McKane, Luckock, and Bray, Phys. Rev. A 41, 644 (1990)] is used to calculate, in the weak-noise limit, the rate of escape Γ of a particle over a one-dimensional potential barrier, for exponentially correlated noise 〈ξ(t)ξ(t’)〉 =(D/τ)exp{-‖t-t’‖/τ}. For small D, a steepest-descent evaluation of the appropriate path integral yields Γ∼exp(-S/D), where S is the ‘‘action’’ associated with the dominant (‘‘instanton’’) path. Analytical results for S are obtained for small and large τ, and (essentially exact) numerical results for intermediate τ. The stationary joint probability density for the position and velocity of the particle is also calculated for small D: it has the form $P_{\mathrm{st}}$ (x,ẋ)∼exp[-S(x,ẋ)/D]. Results are presented for the marginal probability density $P_{\mathrm{st}(\mathrm{x}}$) for the position of the particle.