ASYMPTOTIC ANALYSIS OF FIRST-PASSAGE PROBLEMS

The response of stochastically-force dynamical systems is analyzed in the limit of vanishing noise strength epsilon. We predict asymptotic e xpressions for the stationary response probability density function (p .d.f.) and for the probability of first-passage of the response to the boundary of a domain in state space. The analysis is limited to Gauss ian white noise type perturbations and to domains D in the phase plane ''attracted'' to an equilibrium point O of the system: all unperturbe d trajectories enter D and converge asymptotically to O. In the first stage, the p.d.F. is expressed in terms of an asymptotic WKB form wexp (-Psi/epsilon) where the ''quasi-potential'' Psi can be readily determ ined numerically by a method of ''rays''. A domain of reliability D ma y then be defined as one bounded by a given contour of quasi-potential , since the latter is a Lyapunov function of the deterministic system. In a second stage, the probability of first-passage is determined in terms of the mean first-passage time to the boundary delta D. The latt er is found in a singular perturbation solution devised by Matkowsky a nd Schuss [SIAM J. Appl. Math. 33, 365 (1977)] in terms of the values reached on delta D by Psi, w, and by the deterministic force vector. S everal examples demonstrate the validity and usefulness of this approa ch. Copyright (C) 1996 Elsevier Science Ltd.