  Consider the Allen-Cahn equation on the $d$-dimensional torus, $d=2,3$, in
the sharp interface limit. As it is well known, the limiting dynamics is
described by the motion by mean curvature of the interface between the two
stable phases. Here, we analyze a stochastic perturbation of the Allen-Cahn
equation and describe its large deviation asymptotics in a joint sharp
interface and small noise limit. Relying on previous results on the variational
convergence of the action functional, we prove the large deviation upper bound.
The corresponding rate function is finite only when there exists a time
evolving interface of codimension one between the two stable phases. The zero
level set of this rate function is given by the evolution by mean curvature in
the sense of Brakke. Finally, the rate function can be written in terms of the
sum of two non-negative quantities: the first measures how much the velocity of
the interface deviates from its mean curvature, while the second is due to the
possible occurrence of nucleation events.
