## Abstract

The invariant measure of a one-dimensional Allen-Cahn equation with an additive space-time white noise is studied. This measure is absolutely continuous with respect to a Brownian bridge with a density that can be interpreted as a potential energy term. We consider the sharp interface limit in this setup. In the right scaling this corresponds to a Gibbs-type measure on a growing interval with decreasing temperature. Our main result is that in the limit we still see exponential convergence towards a curve of minimizers of the energy if the interval does not grow too fast. In the original scaling, the measure is concentrated on configurations with precisely one jump.

Original language | English |
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Pages (from-to) | 1071-1109 |

Number of pages | 39 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 63 |

Issue number | 8 |

DOIs | |

Publication status | Published - 1 Aug 2010 |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics