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We study the sharp interface limit for the 1D stochastic Allen-Cahn equation, and extend earlier work by Funaki to the full small noise regime. The main new idea is the construction of a series of functional correctors, which are designed to recursively cancel potential divergences. In addition, in order to show these correctors are well-behaved, we develop a systematic decomposition of functional derivatives of the deterministic Allen-Cahn flow of all orders. This decomposition is of its own interest, and may be useful in other situations as well.
Xu et al. (Thu,) studied this question.