  The influence of small random perturbations on a deterministic dynamical
system with a locally stable equilibrium is considered. The perturbed system is
described by the It\^{o} stochastic differential equation. It is assumed that
the noise does not vanish at the equilibrium. In this case the trajectories of
the stochastic system may leave any bounded domain with probability one. We
analyze the stochastic stability of the equilibrium under a persistent
perturbation by white noise on an asymptotically long time interval $0\leq
t\leq \mathcal O(\mu^{-N})$, where $0<\mu\ll 1$ is a perturbation parameter.
