  The phase space for a periodically driven nonlinear oscillator consists of
many resonance zones. Let the strength of periodic excitation and the strength
of the damping be indexed by a small parameter $\varepsilon$. It is well known
that, as $\varepsilon \to 0$, the measure of the set of initial conditions
which lead to 'capture in a resonance zone' goes to zero. In this paper we
study the effect of weak noise on the escape from a resonance zone.
