  The Allen-Cahn action functional is related to the probability of rare events
in the stochastically perturbed Allen-Cahn equation. Formal calculations
suggest a reduced action functional in the sharp interface limit. We prove in
two and three space dimensions the corresponding lower bound. One difficulty is
that diffuse interfaces may collapse in the limit. We therefore consider the
limit of diffuse surface area measures and introduce a generalized velocity and
generalized reduced action functional in a class of evolving measures. As a
corollary we obtain the Gamma convergence of the action functional in a class
of regularly evolving hypersurfaces.
