Asymptotic analysis of first-passage problems

Abstract

The response of stochastically-forced dynamical systems is analyzed in the limit of vanishing noise strength ε. We predict asymptotic expressions 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 Gaussian white noise type perturbations and to domains D in the phase plane “attracted” to an equilibrium point O of the system: all unperturbed 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( −Ψ ε ) where the “quasi-potential” Ψ can be readily determined numerically by a method of “rays”. A domain of reliability D may 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 ∂ D. The latter is found in a singular perturbation solution devised by Matkowsky and Schuss [ SIAM. Appl. Math. 33, 365 (1977)] in terms of the values reached on ∂ D by Ψ, w and by the deterministic force vector. Several examples demonstrate the validity and usefulness of this approach.