The problem of "exit against a flow" for dynamical systemssubject to small Gaussian white noise excitation is studied.Here the word "flow" refers to the behavior in phase space ofthe unperturbed system's state variables. "Exit against a flow"occurs if a perturbation causes the phase point to leave a phasespace region within which it would normally be confined. Inparticular, there are two components of the problem of exitagainst a flow:
i) the mean exit time
ii) the phase-space distribution of exit locations.
When the noise perturbing the dynamical systems is small, thesolution of each component of the problem of exit against a flowis, in general, the solution of a singularly perturbed, degenerateelliptic-parabolic boundary value problem.
Singular perturbation techniques are used to express theasymptotic solution in terms of an unknown parameter. The unknownparameter is determined using the solution of the adjointboundary value problem.
The problem of exit against a flow for several dynamicalsystems of physical interest is considered, and the mean exittimes and distributions of exit positions are calculated. The systemsare then simulated numerically, using Monte Carlo techniques,in order to determine the validity of the asymptotic solutions.
