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

Abstract

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 \ensuremath{\Gamma} of a particle over a one-dimensional potential barrier, for exponentially correlated noise 〈\ensuremath{\xi}(t)\ensuremath{\xi}(t')〉 =(D/\ensuremath{\tau})exp{-\ensuremath{\Vert}t-t'\ensuremath{\Vert}/\ensuremath{\tau}}. For small D, a steepest-descent evaluation of the appropriate path integral yields \ensuremath{\Gamma}\ensuremath{\sim}exp(-S/D), where S is the ``action'' associated with the dominant (``instanton'') path. Analytical results for S are obtained for small and large \ensuremath{\tau}, and (essentially exact) numerical results for intermediate \ensuremath{\tau}. 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,x\ifmmode \dot{}\else \.{}\fi{})\ensuremath{\sim}exp[-S(x,x\ifmmode \dot{}\else \.{}\fi{})/D]. Results are presented for the marginal probability density ${P}_{\mathrm{st}(\mathrm{x}}$) for the position of the particle.