Some results on the problem of exit from a domain

Abstract

The problem of exit from a domain of attraction of a stable equilibrium point in the presence of small noise is considered for a class of two-dimensional systems. It is shown that for these systems, the exit measure is ‘skewed’ in the sense that if S denotes the saddle point in the quasipotential towards which the exit measure collapses as the noise intensity goes to zero, then there exists an ε dependent neighborhood Δ of S such that lim P(exitinΔ)/∣Δ∣=0. Thus, the most probable exit point is not S but is rather skewed aside by εγ for some γ. The behaviour of such skewness, which was predicted by asymptotic expansions, depends on the ratio of normal to tangential forces around the saddle point.