The sharp-interface limit of the action functional for Allen-Cahn in one space dimension

We analyze the sharp-interface limit of the action minimization problem for the stochastically perturbed Allen-Cahn equation in one space dimension. The action is a deterministic functional which is linked to the behavior of the stochastic process in the small noise limit. Pre- viously, heuristic arguments and numerical results have suggested that the limiting action should \count" two competing costs: the cost to nucleate interfaces and the cost to propagate them. In addition, con- structions have been used to derive an upper bound for the minimal action which was proved optimal on the level of scaling. In this paper, we prove that for d = 1, the upper bound achieved by the constructions is in fact sharp. Furthermore, we derive a lower bound for the func- tional itself, which is in agreement with the heuristic picture. To do so, we characterize the sharp-interface limit of the space-time energy mea- sures. The proof relies on an extension of earlier results for the related elliptic problem