Action minimization and sharp-interface limits for the stochastic allen-cahn equation

We study the action minimization problem that is formally associated to phase transformation in the stochastically perturbed Allen-Cahn equation. The sharpinterface limit is related to (but different from) the sharp-interface limits of the related energy functional and deterministic gradient flows. In the sharp-interface limit of the action minimization problem, we find distinct "most likely switching pathways," depending on the relative costs of nucleation and propagation of interfaces. This competition is captured by the limit of the action functional, which we derive formally and propose as the natural candidate for the Γ-limit. Guided by the reduced functional, we prove upper and lower bounds for the minimal action that agree on the level of scaling.